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New Strings for Old VenezianoAmplitudes III. Symplectic Treatment

A.L. Kholodenko375 H.L.Hunter Laboratories, Clemson University, Clemson,SC 29634-0973, USA

AbstractA d−dimensional rational polytope P is a polytope whose vertices are lo-

cated at the nodes of Zd lattice. Consider the number∣

∣kP ∩ Zd∣

∣ of points insidethe inflated P with coefficient of inflation k (k = 1, 2, 3, ...). The Ehrhart poly-nomial of P counts the number of such lattice points inside the inflated P and(may be) at its faces (including vertices). In Part I [ JGP 55 (2005) 50] of ourfour parts work we noticed that Veneziano amplitude is just the Laplace trans-form of the generating function (considered as a partition function in the senceof statistical mechanics) for the Ehrhart polynomial for the regular inflatedsimplex obtained as deformation retract of the Fermat (hyper) surface livingin the complex projective space. This observation is sufficient for developmentof new symplectic (this work) and supersymmetric (Part II) physical modelsreproducing the Veneziano (and Veneziano-like) amplitudes. General ideas (e.g.those related to the properties of Ehrhart polynomials) are illustrated by sim-ple practical examples (e.g. use of mirror symmetry for explanation of availableexperimental data on ππ scattering , etc.) worked out in some detail. Obtainedfinal results are in formal accord with those earlier obtained by Vergne [PNAS93 (1996)14238 ].

MSC: 81T30; 13A50; 20F55; 20G45; 50B20

Subj Class.: String theory; Torus actions on symplectic manifolds; Combi-natorics of polytopes; Weyl character formula

Keywords : Veneziano and Veneziano-like amplitudes; Ehrhart polynomial;Toric varieties and fans; Reflexive polytopes; Mirror symmetry; Coadjoint or-bits; Moment maps; Duistermaat-Heckman formula; Khovanskii-Pukhlikov cor-respondence

1

1 Introduction

1.1 Connection with earlier work

In our earlier works, Refs.[1,2], which we shall call Part I and II1, we initiateddevelopment of new formalism reproducing both the Veneziano and Veneziano-like (tachyon-free) amplitudes and models generating these amplitudes. In par-ticular, in Part II we discussed one of such models. Contrary to traditionaltreatments, we demonstrated that our model is supersymmetric and finite-dimensional. This result was obtained with help of the theory of invariantsof pseudo-reflection groups. The partition function, Eq.(II,6.10), for this modelis given by the Poincare′ polynomial

P ((S(V ) ⊗ E(V ))G; z) =

n∏

i=1

1 − zq+i

1 − zi. (1.1)

In the limit: z → 1, the above result is reduced to

P ((S(V ) ⊗ E(V ))G; z = 1) =(q + 1)(q + 2) · · · (q + n)

n!≡ p(q, n). (1.2)

which is Eq.(II,6.11). The detailed combinatorial explanation of these resultswas given already in Part II. In this work, to avoid repetitions, we would liketo extend such an explanation having in mind development of the symplecticmodel reproducing Veneziano amplitudes.

Steps toward designing of such a model were made already in Part I where itwas noticed that the unsymmetrized Veneziano amplitude is obtainable as theLaplace transform of the partition function

P (q, t) =

∞∑

n=0

p(q, n)tn (1.3)

where p(q, n) is the same as in Eq.(1.2). In Ref.[3] Vergne demonstrated (withoutreference to string theory or Veneziano amplitudes) that such partition functionhas both symplectic and quantum mechanical meaning : the quantity p(q, n)is dimension of the quantum Hilbert space associated (through the coadjointorbit method) with the classical system made out of finite number of harmonicoscillators living on a specially designed symplectic manifold.

In this work using different arguments we reobtain her final results. Our useof different arguments is motivated by our desire to demonstrate connectionsbetween the formalism developed in this paper and that already in use in the

1In referring to the results of these papers we shall use notations like Eq.(II. 5.10), etc.

2

mathematical physics literature. More importantly, the treatment presentedbelow complements that developed earlier in Parts I and II.

In Part II, following work by Lerche et al [4], we adopted the idea that anykind of one variable Poincare′ polynomial (actually, up to a constant) can beinterpreted as the Weyl character formula. Since, according to Part II, bothEq.s (1.1) and (1.3) are Poincare′ polynomials, their interpretation in terms ofthe Weil character formula provides major ingredient toward reconstruction ofthe Veneziano amplitudes from the underlying quantum mechanical partitionfunction (the Weyl character formula). Going into opposite direction, suchamplitudes acquire some topological meaning to be further illuminated in thiswork. Direct link between topology (the Poincare′ polynomials) and quantummechanics (the Weyl character formula) is certainly not limited to its use onlyfor the Veneziano amplitudes and is of independent interest. In view of this, inthe next subsection we would like to provide simple arguments (different fromthose in the work by Lerche et al) explaining why this is so.

1.2 A motivating example

Consider a finite geometric progression of the type

F(c,m) =

m∑

l=−m

expcl = exp−cm∞∑

l=0

expcl + expcm0∑

l=−∞

expcl

= exp−cm 1

1− expc + expcm 1

1 − exp−c

= exp−cm[

expc(2m+ 1) − 1

expc − 1

]

. (1.4)

The reason for displaying the intermediate steps will become apparent shortly.First, however, we would like to consider the limit : c→ 0+ of F(c,m). Clearly,it is given by F(0,m) = 2m+1. The number 2m+1 equals to the number of in-teger points in the segment [−m,m] including boundary points. It is convenientto rewrite the above result in terms of x = expc. We shall write formallyF(x,m) instead of F(c,m) from now on. Using these notations, let us considerthe related function,

F(x,m) = (−1)F(1

x,−m). (1.5)

Such type of relation (the Ehrhart-Macdonald reciprocity law) is characteristicfor the Ehrhart polynomial for the rational polytopes to be defined in the nextsubsection. In Ref.[5] Stanley provides many applications of this reciprocitylaw. In our case, we obtain explicitly,

F(x,m) = (−1)x−(−2m+1) − 1

x−1 − 1xm. (1.6)

In the limit x → 1 + 0+ we obtain : F(1,m) = 2m − 1. The number 2m − 1is equal to the number of integer points strictly inside the segment [−m,m].

3

These, seemingly trivial, results can be broadly generalized. First, we replacex by x. Next, we replace the summation sign in the left hand side of Eq.(1.4)by the multiple summation, etc. Thus obtained function F(x,m) in the limitxi → 1 + 0+, i = 1 − d, produces the anticipated result:

F(1,m) = (2m+ 1)d (1.7)

for number of points inside and at the edges of the d dimensional cube in Eu-clidean space Rd. Accordingly, for the number of points strictly inside the cubewe obtain: F(1,m) = (2m− 1)d. The rationale for describing this limiting pro-cedure is caused by its connection with our earlier result, Eq.(1.2). To explainthis we need to extend our simple results in order to describe analogous situa-tion for arbitrary centrally symmetric polytope. This is accomplished in severalsteps. We begin with some definitions.

Definition 1.1. A subset of Rn is a polytope (or polyhedron) P if there isa r × d matrix M (with r ≤ d) and a vector b ∈ Rd such that

P = x ∈ Rd | Mx ≤ b. (1.8)

Definition 1.2. Provided that the Euclidean d-dimensional scalar productis given by

< x · y >=

d∑

i=1

xiyi (1.9)

2a rational ( respectively, integral) polytope (or polyhedron) P is defined by theset

P = x ∈ Rd |< ai · x >≤ βi , i = 1, ..., r (1.10)

where ai ∈ Qnand βi ∈ Q for i = 1, ..., r (respectively, ai ∈ Zn and βi ∈ Z fori = 1, ..., r).

Let VertP denote the vertex set of the rational polytope, in the case con-sidered thus far, the d−dimensional cube. Let uv

1, ..., uvd be the orthogonal

basis (not necessarily of unit length) made of the highest weight vectors of theWeyl-Coxeter reflection group Bd appropriate for the cubic symmetry3.Thesevectors are oriented along the positive semi axes with respect to the center ofsymmetry of (hyper)cube. When parallel translated to the edges ending at par-ticular hypercube vertex v, they can point either in or out of this vertex. Interms of these notations, the d-dimensional version of Eq.(1.4) can be rewrittennow as follows

∑

x∈P∩Zd

exp< c · x > =∑

v∈V ertP

exp< c · v >[

d∏

i=1

(1 − exp−ciuvi )]−1

.

(1.11)Correctness of this equation can be readily checked by considering special casesof a segment, square, cube, etc. The result, Eq.(1.11), obtained for the polytope

2So that x lives in space dual to that for y.3For a brief guide to the Weyl-Coxeter reflection groups, please, see Appendix to Part II

4

of cubic symmetry can be extended to the arbitrary convex centrally symmetricpolytope as we shall demonstrate below. This fact allows us to investigateproperties of more complex polytopes with help of polytopes of cubic symmetry.Moreover, we shall argue below that the r.h.s. of Eq.(1.11) is mathematicallyequivalent to the r.h.s. of Eq.(1.1). Because of this, the limiting procedurec → 0+ producing the number of points inside (and at the boundaries) of thepolyhedron P in the l.h.s. of Eq.(1.11) is of the same nature as the limitingprocedure:z → 1 in Eq.(1.2) where, as result of this procedure, the r.h.s ofEq.(1.2) produces the number of lattice points for the inflated (with inflationcoefficient q) rational simplex of dimension n ”living” in Zn lattice.

Remark 1.3. For an arbitrary convex polytope the above formula, Eq.(1.11),was obtained (seemingly independently) in many different contexts. For in-stance, in the context of discrete and computational geometry it is attributedto Brion [6]. In view of Eq.(1.5), it can as well be attributed to Ehrhart, Stanley[5] and to many others. In fact, for the case of centrally symmetric polytopesthis formula is just a special case of the Weyl’s character formula. This will bedemonstrated below, in Section 2.

There are many paths to arrive at final results of this paper, i.e. to reob-tain the results of Vergne [3], and to use them for construction of new modelsreproducing the Veneziano and Veneziano-like amplitudes. They include, forinstance, the algebro-geometric, symplectic, group-theoretic, combinatorial, su-persymmetric, etc. pathways to reach the same destination. In our opinion,the most direct way to arrive at final results is combinatorial. Although it willbe discussed at length in in Part IV from yet another perspective, in this workwe present some essentials needed for their immediate use in the rest of thepaper. In particular, we would like to discuss now some facts about the Ehrhartpolynomials having in mind their uses in high energy physics.

1.3 Ehrhart polynomials, mirror symmetry and the ex-

tended Veneziano amplitudes

In the previous subsection we introduced Eq.(1.5). It characterizes the Ehrhartpolynomial. It is important to realize that p(q, n) in Eq.(1.2) already is anexample of the Ehrhart polynomial. Evidently, Eq.(1.2) can be written formallyas

p(q, n) = anqn + an−1q

n−1 + · · · + a0. (1.12)

In Ref.[7] it is argued that for any convex rational polytope P the Ehrhartpolynomial can be written as

|qP ∩ Zn| = P(q, n) = an(P)qn + an−1(P)qn−1 + · · · + a0(P) (1.13)

with coefficients a0, ..., an specific for a given polytope P . Nevertheless, irrespec-tive to the type of polytope P , it is known that a0 = 1 and an = V olP ,whereV olP is the Euclidean volume of the polytope. To calculate the remaining coeffi-cients of this polynomial explicitly for an arbitrary convex polytope is a difficult

5

task in general. Such task was accomplished rather recently in Ref.[8].The au-thors of [8] recognized that in order to obtain the remaining coefficients it isuseful to calculate the generating function for the Ehrhart polynomial. In ourcase this function is given by Eq.(1.3). From Eq.(I,1.22) we already know thatformally this is the partition function for the unsymmetrized Veneziano ampli-tude. In view of our Eq.(1.1) taken from Part II, we also know that it can be alsolooked upon as the partition function for theVeneziano amplitudes. Hence, nowwe would like to explain how Eq.s (1.1) and (1.3) are related to each other fromthe point of view of commutative algebra and combinatorics of polytopes.Bydoing so some useful physical information will be obtained as well.

Long before the results of Ref.[8] were published, it was known [9] thatthe generating function for the Ehrhart polynomial of P can be written in thefollowing universal form

F(P , x) =

∞∑

q=0

P(q, n)xq =h0(P) + h1(P)x+ · · · + hn(P)xn

(1 − x)n+1. (1.14)

For the Veneziano partition function all coefficients, except h0(P), are zero and,of course, h0(P) = 1 [10]. This can be easily understood in view of Eq.(I.1.20).We brought to our readers attention the above general result in view of our taskof comparing Eq.s(1.1) and (1.3). and of possibly generalizing the Venezianoamplitudes and the partition functions associated with them.

In practical applications it should be noted that the combinatorial factorp(q, n),Eq.(1.2), representing the number of points in the inflated simplex P(with coefficient of inflation q) whose vertex set VertP belongs to Zn lattice canbe formally written in several equivalent ways. In particular, as we’ve mentionedin Part II,

p(q, n) =(q + n)!

q!n!=

(q + 1) · · · (q + n)

n!=

(n+ 1) · · · (n+ q)

q!. (1.15)

This fact has some physical significance. For instance, in particle physics lit-erature, e.g. see Refs.[11,12], the third option is commonly used. Let us recallhow this happens. One is looking for an expansion of the factor (1 − x)−α(t)−1

under the integral of beta function as explained in Part I. Looking at Eq.(1.14)one realizes that the Mandelstam variable α(t) plays a role of dimensionality ofZ- lattice. Hence, we have to identify it with n in the second option providedby Eq.(1.15). This is not the way such an identification is done in physics liter-ature where, in fact, the third option provided by Eq.(1.15) is commonly usedwith n = α(t) effectively being the inflation factor while q effectively being thedimensionality of the lattice.4 A quick look at Eq.s(1.3) and (1.14) shows thatunder such circumstances the generating function for the Ehrhart polynomialand that for the Veneziano amplitude are formally not the same: in the first(mathematical) case one is dealing with lattices of fixed dimensionality and is

4We have to warn our readers that, to our knowledge, nowhere in physics literature suchcombinatorial terminology is being used.

6

considering summation over various inflation factors at the same time, while inthe third (physical) case, one is dealing with the fixed inflation factor n = α(t)while summing over lattices of different dimensionalities. Such arguments aresuperficial however in view of Eq.s(I.1.20) and (1.14) above. Using these equa-tions it is clear that correct agreement between Eq.s(1.3) and (1.14) can bereached if one is using P(q, n) = p(q, n) with the second (i.e.mathematical)option offered by Eq.(1.15). By doing so no changes in the pole locations forthe Veneziano amplitude occur. Moreover, for a given pole the second andthe third option in Eq.(1.15) produce exactly the same contributions into theresidue thus making them physically indistinguishable. Nevertheless, our choiceof the mathematically meanigful interpretation of the Veneziano amplitude asthe Laplace transform of the Ehrhart polynomial generating function providesone of the major reasons for development of the formalism of Parts I-throughIV. In particular, it allows us to think about possible generalizations of theVeneziano amplitude using generating functions for the Ehrhart polynomialsfor polytopes of other types. As it is demonstrated by Stanley [9,13], bothEq.(1.1) and Eq.(1.14) have group- invariant meaning as Poincare′ polynomials:Eq.(1.14) is associated with the Poincare′ polynomial for the so called Stanley-Reisner polynomial ring while Eq.(1.1) is the Poincare polynomial for the socalled Gorenstein ring. Naturally, these two rings are interrelated thus provid-ing the desired connection between Eq.s(1.1) and (1.14). For the sake of space,we refer our readers to the original works by Stanley [9,13] where all mathe-matical details can be found. At the same time, we have supplied sufficientinformation in order to discuss some physical applications. In particular, fol-lowing Batyrev, Ref.[14, p.392 ], and Hibi, Ref.[15], we would like to discuss thereflexive (polar(or dual)) polytopes playing major role in calculations involvingmirror symmetry. To those of our readers who are familiar with some basicfacts of solid state physics [16] the concept of a dual (or polar) polytope shouldlook quite familiar since it is completely analogous to that for the reciprocallattice. Both direct and reciprocal lattices are used rutinely in calculations re-lated to physical properties of crysalline solids. The requirement that physicalobservables should remain the same irrespective to what lattice is used in com-putations is completely natural. Not surprisingly, such a requirement formallycoincides with that used in the high energy physics. In the paper by Greeneand Plesser, Ref.[17, p.26], one finds the following statement: ”Thus, we havedemonstrated that two topologically distinct Calabi-Yau manifolds M and M ′

give rise to the same conformal field theory. Furthermore, although our argu-ment has been based only at one point in the respective moduli spaces MM

and MM ′ of M and M ′(namely the point which has a minimal model inter-pretation and hence respects the symmetries by which we have orbifolded) theresult necessarily extends to all of MM and MM ′ ”. In parts I and II we arguedthat, in view of the Veneziano condition, there is a significant difference betweencalculations of observables (amplitudes) of high energy physics and those in con-formal field theories. This difference is analogous to the difference between thepoint group symmetries in the liquid/gas phases and translational symmetries ofsolid phases. Hence, extension of the Veneziano amplitudes with help of general

7

result, Eq.(1.14), (which is essentially equivalent to accounting for the mirrorsymmetry) requires some explanations. We would like to to provide a sketch ofthese explanations now5.

To this purpose we need to introduce several definitions first. We begin with

Definition 1.4. For any convex polytope P the dual polytope P∗ is definedby

P∗ = x ∈(

Rd)∗ |< a · x >≤ 1,a ∈ P (1.16)

Although in algebraic geometry of toric varieties the inequality < a · x >≤ 1sometimes is replaced by < a · x >≥ −1 [10], we shall use Eq.(1.16) to be inaccord with Hibi, Ref.[15]. According to this reference, if P is rational, then P∗

is also rational. Hovever, P∗ is not necessarily integral even if P is integral.Thisfact is profoundly important since the result, Eq.(1.14), is valid for the integralpolytopes only.The question arises : under what conditions is the dual polytopeP∗ inegral? The aswer is given by the following

Theorem 1.5.(Hibi, Ref.[15]). The dual polytope P∗ is integral if and onlyif

F(P , x−1) = (−1)d+1xF(P , x) (1.17)

where the generating function F(P , x) is defined earlier by Eq.(1.14).By combining Eq.s(1.3) and (1.14) we obtain for the standard Veneziano

amplitude the following result:

F(P , x) =

(

1

1 − x

)d+1

. (1.18)

Using it in Eq.(1.17) produces:

F(P , x−1) =(−1)d+1

(1 − x)d+1xd+1 = (−1)d+1xd+1F(P , x). (1.19)

This result indicates that scattering processes described by the standard Venezianoamplitudes do not involve any mirror symmetry since, as it is well known, Refs[14,18], in order for such a symmetry to take place the dual polytope P∗ mustbe integral. The question arises: Can these amplitudes be modified with helpof Eq.(1.14) so that mirror symmetry can be in principle observed in nature?To answer this question, let us assume that, indeed, Eq.(1.14) can be used forsuch a modification. In this case we obtain

F(P , x−1) = (−1)d+1xF(P , x) (1.20)

5To keeep focus of our readers on major issues of this paper, we supress to the absolute min-imum the discussion connecting our results to experiment. We plan to discuss this connectionthoroughly in a separate publication.

8

porovided that hn−i = hi in Eq.(1.14). But this is surely the case in view of thefact that these are the Dehn-Sommerville equations, Ref.[10, p.16]. Hence, atthis stage of our discussion, it looks like generalization of the Veneziano ampli-tudes which takes into account mirror symmetry is possible from the mathemat-ical standpoint. Mathematical arguments themself are not sufficient howeverfor such generalization. This is so because of the following chain of arguments.

Already in the original paper by Veneziano, Ref.[19, p.195], it was noticedthat the amplitude he defined originally is not unique. Following Ref.[20, p.100],we notice that beta function B(−α(s),−α(t)) in Veneziano amplitude can bereplaced by

B(m− α(s), n− α(t)) (1.21)

for any integers m,n ≥ 1, and similarly for s, u and t, u terms. According toRef. [20], ”Any function which can be presented as linear combination of termslike (1.21) satisfies the finite energy sum rules, so there is no constraint onthe resonance parameters unless additional assumptions are made.” The mirrorsymmetry arguments presented above are such additional assumptions. To usethem wisely, we still need to make several remarks. Experimentally, the linearcombination of terms in the form given by Eq.(1.21) should show up in the formof daughter (or satellite) Regge trajectories as explained inRef.[20,21]. But,according to Frampton, Ref.[22], such daughter trajectories should be presenteven for the standard (that is non generalized!) Veneziano amplitudes. This isso because of the following arguments. In accord with analysis made in Part I,the unsymmetrized Veneziano amplitude can be presented as

V (s, t) =

∞∑

n=0

p(α(t), n)1

n− α(s). (1.22)

For a given n the polynomial p(α(t), n) is an n − th degree polynomial in α(t)(for high enough energies in t). Since in the Regge theory n represents the totalspin, according to the rules of quantum mechanics, in addition to particles withspin n there should (could) be particles with spins n − 1, n − 2, ..., 1, 0.Theseparticles should be visible (in principle) at the parallel (daughter or satel-lite) trajectories all lying below the leading (with spin n) Regge trajectory.While the leading trajectory has α(t)n as its residue the daugher trajectories

have (α(t))n−1

, (α(t))n−2

,etc.as their residues. Unfortunately, in addition, thecountable infinity of satellite (daughter) trajectories can originate if the massesof colliding particles are not the same, Ref.[20, p. 40].The linear combination ofterms given by Eq.(1.21) can account in principle for such phenomena. Follow-ing Frampton, Ref.[22], we need to take into account that the linear combinationof terms in Eq.(1.21) can be replaced (quite rigorously) by the combination ofterms of the form

B(n− α(s), n− α(t)) (1.23)

with n ≥ 0.In real life the infinite number of trajectories is never observed how-ever. But several parent-daughter Regge trajectories are being observed quitefrequently, e.g. see Ref.[20, p.41]. If we accept the existence of mirror symmetry

9

these observational facts can be explained quite naturally. To illustrate this, wewould like to consider the simplest case of ππ scattering desribed in Ref.s[12,22].Below the threshold, that is below the collision energies producing more outgo-ing particles than incoming, the unsymmetrized amplitude A(s, t) for such aprocess is known to be

A(s, t) = −g2 Γ(1 − α(s))Γ(1 − α(t))

Γ(1 − α(s) − α(t))

= −g2(1 − α(s) − α(t))B(1 − α(s), 1 − α(t)). (1.24)

This result should be understood as follows. Consider the ”weighted” (unsym-metrized) Veneziano amplitude of the type

A(s, t) =

1∫

0

dxx−α(s)−1(1 − x)−α(t)−1g(x, s, t) (1.25)

where the weight function g(x, s, t) is given by

g(x, s, t) =1

2g2[(1 − x)α(s) + xα(t)]. (1.26a)

Alternatively, the same result, Eq.(1.24), is obtained if one uses instead theweight function

g(x, s, t) = g2xα(t) (1.26b)

Consider now a special case of Eq.(1.14): n = 2. For such case we obtain

F(P , x) =

∞∑

q=0

P(q, 1)xq =h0(P) + h1(P)x

(1 − x)1+1(1.27)

so that Eq.(1.20) holds thus indicating presence of mirror symmetry. At thispoint our readers might notice that, actually, for this symmetry to take placeone should consider instead of amplitude A(s, t) the following combination

A(s, t) = −g2 Γ(1 − α(s))Γ(1 − α(t))

Γ(1 − α(s) − α(t))+ g2 Γ(−α(s))Γ(−α(t))

Γ(−α(s) − α(t))

= −g2B(1 − α(s), 1 − α(t)) + g2B(−α(s),−α(t)) (1.27)

Such a combination produces first two terms (with correct signs) of the infiniteseries as proposed by Mandelstam, Eq.(15), Ref.[23]. The comparison with ex-periment displayed in Fig.6.2 (a), Ref[22, p. 321] is quite satisfactory producingone parent and one daughter trajectory. These are also displayed in Ref.[20,p.41] for the ”rho family”. It should be noted that in the present case thephase factors (eliminating tachyons) discussed in Part I are not displayed sincein Ref.[24] and, therefore also in this work, we provide alternative explanationwhy tachyons should be excluded from consideration.

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1.4 Organization of the rest of the paper

In Section 2 using the concept of a zonotope we prove that, indeed, Eq.(1.11)(and, hence, Eq.s (1.1) and (1.3)) is a special case of the Weyl character for-mula. In arriving at this result we employ some information about the Ruelle’sdynamical transfer operator and earlier results by Atiyah and Bott on Lefshetz-type fixed point formula for the elliptic complexes. Section 3 along with resultsof Appendix (Part II) provides necessary mathematical background to be usedin Section 4. It includes some relevant facts from the theory of toric varieties,algebraic groups, semisimple Lie groups and assiciated with them Lie algebras,flag decompositions, etc. With help of this information in Section 4 major phys-ical applications are developed culminating in the exact symplectic solution ofthe Veneziano model. Connection between the symplectic and supersymmetricformalisms was noticed and developed in the calssical paper by Atiyah and Bott,Ref.[25], whose work had been inspired by earlier result by Witten, Ref. [26],on supersymmetry and Morse theory. Thus, in view of Ref.[25], the results ofPart II and III become interrelated. The final results obtained in this work areare in formal accord with those obtained earlier by Vergne, Ref.[3], by othermethods.

2 From geometric progression to Weyl charac-

ter formula

2.1 From p-cubes to d-polytopes via zonotope construc-

tion

In Section 1 we have obtained Eq.(1.11). In one dimensional case it is formallyreduced to a simple geometric progression formula. The result, Eq.(1.11), isobtained for the rational (or integral) polytope of cubic symmetry. In thissubsection we generalize this result to obtain similar results for rational centrallysymmetric polytopes whose symmetry is other than cubic. This is possible withhelp of the concept of zonotope. The concept of zonotope is not new. Accordingto Coxeter [27], it belongs to the 19th century Russian crystallographer Fedorov.Nevertheless, this concept has been truly appreciated only relatively recently inconnection with oriented matroids. For the purposes of this work it is sufficientto consider only the most elementary properties of zonotopes. Thus, followingRef.[28], let us consider a p−dimensional cube Cp defined by

Cp = x ∈ Rp,−1 ≤ xi ≤ 1, i = 1 − p (2.1)

and the surjective map π : Rp → Rd . The map is defined via the followingDefinition 2.1. A zonotope Z(V ) is the image of a p−cube, Eq.(2.1), under

11

the affine projection π. Specifically,

Z ≡ Z(V ) = VCp + z = V y + z : y ∈Cp

= x ∈ Rd : x = z +

p∑

i=1

xivi ,−1 ≤ x ≤ 1

for some matrix (vector configuration) V = v1, ...,vp.By construction, such an image is a centrosymmetric d-polytope [28]. Below,

we shall obtain some results for these d-polytopes. By construction, they shouldhold also for p-cubes. In such a way we shall demonstrate that, indeed, Eq.(1.11)can be associated with the Weyl character formula.

2.2 From Ruelle dynamical transfer operator to Atiyah

and Bott Lefschetz-type fixed point formula for the

elliptic complexes

Any classical dynamical system can be thought of as the pair (M, f) with fbeing a map f : M → M from the phase space M to itself. Following Ruelle[29],we consider a map f : M → M and a scalar function (a weight function) g:M → C. Based on these data, the transfer operator L can be defined as follows:

LΦ(x) =∑

y:fy=x

g(y)Φ(y). (2.2)

If L1 and L2 are two such transfer operators associated with some successivemaps f1 , f2 : M → M and weights g1 and g2 then,

(L1L2Φ)(x) =∑

y:f2f1y=x

g2(f1y)g1(y)Φ(y). (2.3)

It is possible to demonstrate that

trL =∑

x∈Fixf

g(x)

|det(1 −Dxf−1(x))| (2.4)

withDxf being derivative of f acting in the tangent space TxM and the graph off is required to be transversal to the diagonal ∆ ⊂ M×M . Eq.(2.4) coincideswith that obtained in the work by Atiyah and Bott [30]6. In connectionwith Eq.(2.4), these authors make several important (for purposes of this work)observations to be discussed now. In Ref.[29] Ruelle essentially uses the sametype of arguments as those by Atiyah and Bott [30]. These are given as follows.Define the local Lefschetz index Lx(f) by

Lx(f) =det(1 −Dxf(x))

|det(1 −Dxf(x))| , (2.5)

6They use Dxf instead of Dxf−1 which makes no difference for the fixed points andinvertible functions. The important (for chaotic dynamics) non invertible case is discussedby Ruelle also but the results are not much different.

12

where x ∈ Fixf . Then define the global Lefschetz index L(f ) by

L(f ) =∑

f(x)=x

Lx(f). (2.6)

Taking into account that det(1−Dxf(x))=d∏

i=1

(1 − αi), where αi are the eigen-

values of the Jacobian matrix, the determinant can be rewritten in the followinguseful form, Ref.[31, p.133],

det(1 −Dxf(x)) =

d∏

i=1

(1 − αi) =

d∑

k=0

(−1)kek(α1, ..., αd), (2.7)

where the elementary symmetric polynomial ek(α1, ..., αd) is defined by

ek(α1, ..., αd) =∑

1≤ i1<···< ik≤d

αi1 · · · αik(2.8)

with ek=0 = 1.With help of these results the local Lefschetz index, Eq.(2.5), canbe rewritten alternatively as follows

Lx(f) =

d∑

k=01

(−1)kek(α1, ..., αd)

|det(1 −Dxf(x))| ≡

d∑

k=0

(−1)ktr(∧kDxf(x))

|det(1 −Dxf(x))| (2.9)

with ∧kdenoting the k-th power of the exterior product. Using this result,Ruelle in Ref.[29] defined additional transfer operator L(k) (analogous to earlierintroduced L) as follows.

trL(k) =∑

x∈Fixf

g(x)tr(∧kDxf(x))

|det(1 −Dxf−1(x))| . (2.10)

In view of Eq.s(2.5)-(2.10), we also obtain,

d∑

k=0

(−1)ktrL(k) =∑

x∈Fixf

g(x)Lx(f). (2.11)

If in the above formulas we replace Fixf by Fixfn, we have to replace trL(k)

by trL(k)n . Next, since

exp(

∞∑

n=1

tr(An)

ntn) = [det(1− tA)]

−1(2.12)

13

it is convenient to combine this result with Eq.(2.10) in order to obtain thefollowing dynamical zeta function:

Z(t) = exp(

∞∑

n=1

tn

n

d∑

k=0

(−1)ktrL(k)n

)

=

d∏

k=0

[

exp(

∞∑

n=1

trL(k)n

ntn)

](−1)k

=d∏

k=0

[

det(1−tL(k))](−1)k+1

. (2.13)

This final result coincides with that obtained by Ruelle as required. Thus ob-tained zeta function possess dynamical, number-theoretic and algebro-geometricinterpretation. Looking at Eq.(1.1), it should be clear that for the appropriatelychosen d and L Eq.(1.1) and (2.13) can be made the same. Moreover, basedon the paper by Atiyah and Bott, Ref.[30], we would like to demonstrate thatEq.(2.4) is actually the same thing as the Weyl’s character formula [32]. Toprove that this is indeed the case is not entirely trivial. In what follows, weshall assume that our readers are familiar with results and notations of Part IIand, especially, with the results of Appendix to Part II. To avoid duplications,we shall use below results on the Weyl-Coxeter reflection groups using notationsfrom this Appendix7

2.3 From Atiyah-Bott-Lefschetz fixed point formula to char-

acter formula by Weyl

We begin with observation that Eq.s(2.4) and (1.11) are equivalent. Becauseof this, it is sufficient to demonstrate that the r.h.s of Eq.(1.11) indeed coin-cides with the Weyl’s character formula. Although Eq.(1.11) (and, especially,Eq.s(2.3) and (2.10)) looks similar to that obtained in the paper by Atiyahand Bott (AB), Ref.[30, part I, p.379], leading to the Weyl character formula,Eq.(5.12) of Ref.[30, part II8], neither Eq.(1.11) nor Eq.(5.11) of AB paperRef.[30, part II] provide immediate connection with their Eq.(5.12). Hence, thetask now is to restore some missing links.

To this purpose we need to recall some facts from the book by Bourbaki,Ref.[32].These facts are also helpful in the remainder of this paper. In par-ticular, let us consider a finite set of formal symbols e(µ) possessing the samemultiplication properties as the usual exponents9, i.e.

e(µ)e(ν) = e(µ+ ν), [e(µ)]−1

= e(−µ) and e(0) = 1. (2.14)

7To shorten notations we shall write ”Appendix” having in mind the Appendix of Part II.8It should be takent into account that the AB paper aslo has Parts I and II.9In the case of usual exponents it is being assumed that all the properties of formal expo-

nents are transferable to the usual ones.

14

Such defined set of formal exponents is making a free Z module with the basise(µ). Subsequently, we shall require that µ ∈ ∆ with ∆ being the Weyl rootsystem defined in the Appendix. Suppose also that we are given a polynomialring A[X] made of all linear combinations of terms Xn ≡ Xn1

1 ···Xnd

d with ni ∈ Zand Xi being some indeterminates. Then, one can construct another ring A[P]made of linear combinations of elements e(p · n) with p · n =p1n1 + · · ·pdnd.Clearly, the above rings are isomorphic as it was explained in Part II, Section9. Let x =

∑

p∈P

xpe(p) ∈A[P] with P=p1, ..., pd. Then using Eq.(2.14) we

obtain,

x · y =∑

s∈P

xse(s)∑

r∈P

yre(r) =∑

t∈P

zte(t) with

zt =∑

s+r=t

xsyr and, accordingly,

xm =∑

t∈P

zte(t) with zt =∑

s+···+r=t

xs · · · yr, m ∈ N (2.15)

with N being some non negative integer. Introduce now the determinant ofw ∈ W via rule:

det(w) ≡ ε(w) = (−1)l(w), (2.16)

where, again, we use notations from Appendix. If, in addition, we would require

w(e(p)) = e(w(p)), (2.17)

then all elements of the ring A[P] are subdivided into two classes defined by

w(x) = x (invariance) (2.18a)

andw(x) = ε(x) · x (anti invariance). (2.18b)

These classes are very much like subdivision into bosons and fermions in quan-tum mechanics10. All anti invariant elements can be built from the basic antiinvariant element J(x) which, in view of Eq.(2.7), can defined by

J(x) =∑

w∈W

ε(w) · w(x). (2.19)

From the definition of the set P and from Appendix it should be clear that theset P can be identified with the set of reflection elements w of the Weyl groupW . Therefore, for all x ∈ A[P] and w ∈ W we obtain the following chain ofequalities:

w(J(x)) =∑

v∈W

ε(v) · w(v(x)) = ε(w)∑

v∈W

ε(v) · v(x) = ε(w)J(x) (2.20)

10Such analogy is not superficialas we have noticed already in Part II.

15

as required. Accordingly, any anti invariant element x can be written as x =∑

l∈P

xpJ(exp(p)). The denominator of Eq.(1.11), when properly interpreted

with help of results of Appendix, can be associated with J(x). Indeed, withoutloss of generality let us choose the constant c as c = 1, ..., 1. Then, for thefixed v the denominator of Eq.(1.11) can be rewritten as follows:

d∏

i=1

(1 − exp−uvi ) ≡

∏

α∈∆+

(1 − exp(−α)) ≡ d exp(−ρ), (2.21)

where ρ =1

2

∑

α∈∆+

α and

d =∏

α∈∆+

(exp(α

2) − exp(−α

2)). (2.22)

To prove that thus defined d belongs to the anti invariant subset of A[P] is notdifficult. Indeed, consider the action of a reflection ri on d. Taking into accountthat ri(αi) = −αi we obtain,

ri(d) = (exp(−αi

2) − exp(

αi

2))∏

α 6=αi

α∈∆+

(exp(α

2) − exp(−α

2))

= −d ≡ ε(ri)d. (2.23)

Hence, clearly,

d =∑

p∈P

xpJ(exp(p)). (2.24)

Moreover, it can be shown [32], that d = J(exp(ρ)) which, in view of Eq.s (2.21),(2.22), produces identity originally obtained by Weyl :

d exp(−ρ) =∏

α∈∆+

(1 − exp(−α)). (2.25)

Applying reflection w to the above identity while taking into account Eq.s(2.17),(2.23) produces:

∏

α∈∆+

(1 − exp(−w(α))) = exp(−w(ρ)) ε(w) d. (2.26)

The result just obtained is of central importance for the proof of the Weyl’sformula. Indeed, in view of Eq.s (2.17) and (2.26), inserting the identity :

1 =w

winto the sum over the vertices on the r.h.s. of Eq.(1.11) and taking

into account that: a) ε(w) = ±1 so that [ε(w)]−1

= ε(w) , b) actually, the sumover the vertices is the same thing as the sum over the members of the Weyl-Coxeter group (since all vertices of the integral polytope can be obtained by

16

using the appropriate reflections applied to the highest weight vector pointingto a chosen vertex), we obtain the Weyl’s character formula:

trL(λ) =1

d

∑

w∈∆

ε(w) expw(λ+ ρ). (2.27)

It was obtained with help of the results of Appendix, Eq.s(2.4),(2.17) and (2.26).Looking at the l.h.s. of Eq.(1.11) we can replace trL(λ) by the sum in the l.h.s.of Eq.(1.11) if we choose the constant c as before. Doing this is not tooilluminating however as we would like to explain now.

Indeed, since by construction J(x) in Eq.(2.19) is the basic anti invariantelement and the r.h.s of Eq.(2.27) is by design manifestly invariant element ofthe A[P], it is only natural to look for the basic invariant element analogueof J(x). Then, in view of Eq.(2.15) (and discussion preceding this equation),trL(λ) ≡ χ(λ) should be expressible as follows:

χ(λ) =∑

w∈W

nw(λ) e(w). (2.28)

The factor nw(λ) in Eq.(2.28) is known in group theory as the Kostant’s multi-plicity formula [33]. It plays an important role in our work, especially in Section4. To calculate it explicitly, Cartier [34] simplified the original derivation byKostant. In view of simplicity of his arguments, we would like to reproducethem having in mind their later use in the text. Cartier noticed that the de-nominator of the Weyl character formula, Eq.(2.27), can be formally expandedwith help of Eq.(2.25) as follows:

[

exp(ρ)∏

α∈∆+

(1 − exp(−α))

]−1

=∑

w′∈W

P (w′)e(−ρ− w′). (2.29)

By combining Eq.s(2.27),(2.28) and (2.29) we obtain,

∑

w∈W

nw(λ) e(w) =∑

w∈W

ε(w) expw(λ+ ρ)∑

w′∈W

P (w′)e(−ρ− w′). (2.30)

Comparing the l.h.s. with the r.h.s. in the above expression we obtain finallythe Kostant multiplicity formula:

nw(λ) =∑

w′∈W

ε(w′)P (w′(λ+ ρ) − (ρ+ w)). (2.31)

The obtained formula allows us to determine the density of states nw(λ) Pro-vided that the function P is known explicitly, the obtained formula allows us todetermine the factor nw(λ).

It is useful to rewrite these results in physical language. In particular, forany quantum mechanical system, the partition function Ξ can be written as

Ξ =∑

n

gn exp−βEn ≡ tr(exp(−βH)) (2.32)

17

where H is the quantum Hamiltonian of the system, β is the inverse temperatureand gn is the degeneracy factor. Clearly, using Eq.s(2.27), (2.28), one canidentify Ξ with χ(λ). Next we introduce the density of states ρ(E) via

ρ(E) =∑

n

gnδ(E − En). (2.33)

Comparison between Eq.(2.31) and (2.33) suggests that the function P can beidentified with the density of states. Using ρ(E) the partition function Ξ canbe written as the Laplace transform

Ξ(β) =

∞∫

0

dEρ(E) exp−βE. (2.34)

Clearly, Eq.(2.28) is just the discrete analogue of Eq.(2.34) so that it does havea statistical/quantum mechanical interpretation as partition function. Fromcondensed matter physics it is known that all important statistical/quantuminformation is contained in the density of states. Its calculation is of primaryinterest in physics. Evidently, the same is true in the present case.

In the light of results just obtained, we can reinterpret some results fromthe Introduction. In particular, the r.h.s. our Eq.(1.11), when compared withthe r.h.s. of Eq.(5.11) of AB paper, Ref.[30, part II], is not looking the same.We would like to explain that, nevertheless, these expressions are equivalent.For comparison, let us reproduce Eq.(5.11) of AB paper first. Actually, for thispurpose it is more convenient to use the paper by Bott, Ref.[35], (his Eq.(28)).In notations taken from this reference, Eq.(5.11) by AB is written now as follows:

trace Tg =∑

w∈Wα<0

[

λ∏

(1 − α)

]w

. (2.35)

Comparing this result with the r.h.s. of our Eq.(1.11) and taking into accountEq.(2.17), the combination λw in the numerator of Eq.(2.35) is the same thingas expwλ in Eq.(2.27). As for the denominator, Bott uses the same Eq.(2.25)as we do so that it remains to demonstrate that

[

∏

α∈∆+

(1 − exp(−α))

]w

= exp(−w(ρ)) ε(w) d. (2.36)

In view of Eq.(2.25), we need to demonstrate that

[

d exp(−ρ)]w

= exp(−w(ρ)) ε(w) d, (2.37)

i.e. that[

d]w

= ε(w) d. Looking at Eq.(2.23), this requires us to assume that[

d]w

= wd. But, in view of Eq.s(2.17), (2.19) and (2.24), we conclude that

18

this is indeed the case. This proves the fact that Eq.(2.35), that is Eq.(5.11) ofRef.[30], is indeed the same thing as the Weyl’s character formula, Eq.(5.12) ofRef.[30], or, equivalently, Eq.(2.27) above. According to Kac, Ref.[36, p.174],the classical Weyl character formula, Eq.(2.27), is formally valid for both finitedimensional semisimple Lie algebras and infinite dimensional affine Kac-Moodyalgebras. This circumstance and the Proposition A.1. of Appendix play impor-tant motivating role for developments in our work.

Remark 2.2. Although our arguments thus far have been limited onlyto (comparison with) the d− dimensional hypercube, this deficiency is easilycorrectable with help of the concept of zonotope introduced in Section 2.1.Clearly, because of zonotope construction the obtained results remain correctfor any centrally symmetric polytope P . Thus, we have demonstrated thatEq.(1.11) has essentially the same meaning as the Weyl character formula.

Remark 2.3. In view of Eq.s(2.22),(2.35) the denominator d in the Weylcharacter formula, Eq.(2.27), is actually a determinant. This means that thebasic anti invariant J(x) introduced in Eq.(2.19) is determinant. But the r.h.s.of Eq.(2.27) is invariant. This is possible only if the numerator of the Weylcharacter formula is also a determinant. Hence, the Weyl character formula isessentially the ratio of determinants. Since this is surely the case, it impliesthat the Ruele zeta function, Eq.(2.13) is also essentially the Weyl characterformula11. This, in turn, implies that our major result for the Veneziano parti-tion function, Eq.(1.1), is essentially also the Weyl character formula in accordwith Lerche et al [4], where such conclusion was obtained differently.

The fact that Eq.(1.1) can be interpreted as the Weyl character formulashould not be too surprising in view of the fact that the l.h.s. of Eq.(1.1) denotesthe Poincare polynomial which is group invariant. It remains to demonstratethat the torus action group introduced in Part II can be interpreted as the Weyl-Coxeter reflection group. This is done below, in Section 3. In the meantime,we have not exhausted all consequences of the results we have just obtained. Inparticular, if it is true that Eq.(1.11) is the Weyl character formula, then, takinginto account Eq.(2.28), we conclude that Eq.(1.7) is the Kostant multiplicityformula for d-dimensional cube. The rigorous proof of this fact for the arbitraryconvex polytope can be found in Refs. [33, 38, 39]. If this is so, then Eq.(1.2)is also the Kostant multiplicity formula providing the number of points insidethe inflated (with inflation factor q) simplex q∆n (living in Zn lattice) andat its boundaries. These observations allow us to develop symplectic methodsfor reconstruction of the Veneziano partition function to be discussed below inSection 4.

Remark 2.4. From the point of view of algebraic geometry of toric varieties[40-42], the Kostant multiplicity formula has yet another (topological) interpre-tation as the Euler characteristic χ of the projective toric variety associated withthe polytope P . This fact will be discussed in some detail below, in Sections 3.4

11That this is the case for dynamical systems can be deduced, based on our arguments,from the monograph by Feres, Ref.[37]. We shall not develop this line of thought in this paperhaving in mind different goals.

19

and 4. It motivates us to develop symplectic formulation of the Veneziano par-tition function in Section 4 and supersymmetric formulation discussed in PartII. Connections between the Weyl character formula and the Euler character-istic for projective algebraic varieties had been uncovered by Nielsen, Ref.[43],already in 1976. His work is based on still earlier work by Iversen and Nielsen,Ref.[44]. Below we shall argue that, actually, the idea of such connection canbe traced back to still much earlier papers by Hopf, Ref.[45], and Hopf andSamelson, Ref.[46]. In Section 3.4., we provide the topological interpretationof the Kostant multiplicity formula in terms of χ based on ideas of Hopf andSamelson.

3 The torus action and the moment map

3.1 The torus action and the Weyl group

To avoid duplications, in writing this and following subsections we shall assumethat our readers are familiar with our earlier work, Part II. Hence, in this partwe only introduce terminology which is of immediate use. To begin, let usconsider a polynomial

f(z) = f(z1, ..., zn) =∑

i

λizi =

∑

i

λi1....inzi11 ···zin

n , (λi, zim

m ∈ C,1 ≤ m ≤ n).

(3.1)It belongs to the polynomial ring A[z] (essentially isomorphic to earlier intro-duced A[X]) closed under ordinary addition and multiplication. Since now weare using complex numbers (instead of indeterminates as in Section 2.3.) thisallows us to introduce the following

Definition 3.1. An affine algebraic variety V ∈ Cn is the set of zeros ofthe collection of polynomials from the ring A[z].

According to the famous Hilbert’s Nullstellensatz a collection of such poly-nomials is finite and forms the set I(z) := f ∈ A[z], f(z) = 0 of maximalideals usually denoted SpecA[z].

Definition 3.2.The zero set of a single function belonging to I(z) is calledalgebraic hypersurface so that the set I(z) corresponds to the intersection of afinite number of hypersurfaces.

As in Part II, we need to consider the set of Laurent monomials of the typeλzα ≡ λzα1

1 · · ·zαnn . We shall be particularly interested in the monic monomials

for which λ = 1. Such monomials form a closed polynomial subring with respectto usual multiplication and addition. The crucial step forward is to assume thatthe exponent α ∈ Sσ

12. This allows us to define the following mapping

ui := zai (3.2)

12Although the monoid Sσ was defined in Part II, for reader’s convenience it will be redefinedbelow momentarily

20

with ai being one of the generators of the monoid Sσ and z ∈ C. In order to de-fine the monoid Sσ we still need to provide a couple of definitions. In particular,recall that a semi-group S that is a non-empty set with associative operationis called monoid if it is commutative, satisfies cancellation law (i.e.s+x=t+ximplies s=t for all s,t,x∈ S) and has zero element (i.e.s + 0 = s, s ∈ S).Thisallows us to make the following

Definition 3.3. A monoid S is finitely generated if exist a set a1, ..., ak ∈ S,called generators, such that

S = Z≥0a1 + · · · + Z≥0ak. (3.3)

Taking into account this definition, it is clear that the monoid Sσ =σ ∩ Zd forthe rational polyhedral cone σ(e.g. read Part II) is finitely generated.

The mapping given by Eq.(3.2) provides an isomorphism between theadditive group of exponents ai and the multiplicative group of monic Laurentpolynomials. Next, the function φ is considered to be quasi homogenous ofdegree d with exponents l1, ..., ln if

φ(λl1x1, ..., λlnxn) = λdφ(x1, ..., xn), (3.4)

provided that λ ∈ C∗. Applying this result to za ≡ za1

1 ···zann we obtain equation

analogous to Eq.(3.3) for the polyhedral cone:

∑

j

(lj)iaj = di. (3.5)

Clearly, if the index i is numbering different monomials, then the sum di belongsto the monoid Sσ. The same result can be achieved if instead we would considerthe products of the type ul1

1 · · · ulnn and rescale all z′is by the same factor λ.

Eq.(3.5) should be understood as a scalar product with (lj)iliving in the space

dual to a′js . Accordingly, the set of (lj)′

is can be considered as the set of

generators for the dual cone σ∨. Next, in view of Eq.(3.2), let us consider thepolynomials of the type

f(z) =∑

a∈Sσ

λaza =

∑

l

λlul. (3.6)

As before, they form a polynomial ring. The ideal for this ring is constructedbased on the observation that for the fixed di and the assigned set of conegenerators ai there is more than one set of generators for the dual cone. Thisredundancy produces relations of the type

ul11 · · · ulk

k = ul11 · · · ulk

k . (3.7)

If now we require ui ∈ Ci, then it is clear that the above equation belongs tothe ideal I(z) of the above polynomial ring and that Eq.(3.7) represents thehypersurface in accord with the Definitions 3.1 and 3.2. As before, the idealI(z) represents the intersection of these hypersurfaces thus forming the affine

21

toric variety Xσ∨ . The generators u1, ..., uk ∈ Ck are coordinates for Xσ∨ .

They represent the same point in Xσ∨ if and only if ul =ul. Thus formed toricvariety corresponds to just one (dual) cone. A complex algebraic torus T isdefined by the rule: T := (C\0)n =: (C∗)n.It acts on the affine toric varietyXΣ (made out of pieces Xσ∨ with help of a gluing map) according to the rule :T ×XΣ → XΣ, provided that at each affine variety corresponding to the dualcone σ∨ its action is given by

T ×Xσ∨ → Xσ∨ , (t, x) 7→ tx := (ta1x1, ..., takxk). (3.8)

To proceed, we replace temporarily T by the group G acting (multiplica-tively) on the set X via rule: G ×X → X, i.e. (g, x) → gx, provided that forall g, h ∈ G, g(hx) = (gh)x and ex = x for some unit element e of G.

Definition 3.4. The subset Gx := gx | g ∈ X of X is called the orbitof x. The subgroup H := gx = x | g ∈ X of G that fixes x is the isotropygroup. There could be more than one fixed point for the equation gx = x. Allof them are conjugate to each other.

Definition 3.5. A homogenous space for G is the subspace of X on whichG acts without fixed points.

The major step forward can be made by introducing the concept of an alge-braic group [47].

Definition 3.6. A linear algebraic group G is a) an affine algebraic varietyand b) a group in the sense given above, i.e.

µ : G×G→ G ; µ(x, y) = xy (3.9a)

i : G→ G ; ι(x) = x−1. (3.9b)

Remark.3.7. It can be shown, Ref. [48, p.150], that G as a linear algebraicgroup is isomorphic to a closed subgroup of GLn(K) for some n ≥ 1 and anyclosed number field K such as C or p−adic. This fact plays the central role inwhole development presented below.

Consider therefore an action of G on f(z) defined by Eq.(3.6). FollowingStanley, Ref. [13], it can be defined as M f(z) = f(Mz) for some matrix Msuch that M ∈ G. In order for this definition to be compatible with earlier made,Eq.(3.8), we have to assume that the torus T acts diagonally on the vector spacespanned by x1, ..., xn. This means that the isotropy group of the torus is definedby the set of the following equations

taixi = xi . (3.10)

Apart from trivial solutions: xi = 0 and xi = ∞, there are nontrivial solutions:tai = 1 ∀ xi. For integer a′is these are cyclotomic equations whose ai − 1

solutions all lie on the circle S1

(e.g. see PartI, Section 3.1.) This result iseasy to understand since the algebraic torus T has the topological torus as a

22

deformation retract while the topological torus is just a Cartesian product ofcircles.

Next, we notice that Eq.(3.8) still makes sense if some of t−factors arereplaced by 1’s. This means that one should take into account situations whenone, two, etc. t-factors in Eq.(3.8) are replaced by 1’s and account for allpermutations involving such cases. This observation leads to the torus actionson toric subvarieties. It is important that different orbits belong to differentsubvarieties which do not overlap. Thus, by design, XΣ is the disjoint unionof finite number of orbits identified with the subvarieties of XΣ.Under suchcircumstances the vector (x1, ..., xk) forms a basis of k−dimensional vectorspace V so that the vector (x1, ..., xi) , i ≤ k, forms a basis of the subspace Vi .This allows us to introduce a complete flag f0 of subspaces in V via

f0 : 0 = V0 ⊂ V1 ⊂ ... ⊂ Vk = V. (3.11)

Consider now an action of G on f0 . Taking into account the Remark 3.8.,we recognize that effectively G=GLn(K). The matrix representation of thisgroup possess remarkable properties. These are summarized in the followingdefinitions.

Definition 3.8. Given that the set GLn(K) = x ∈ Mn(K)| detx 6= 0with Mn(K) being n × n matrix with entries xi,j ∈ K forms a general lineargroup, the matrix x ∈Mn(K) is a) semisimple (x = xs), if it is diagonalizable,that is ∃g ∈ GLn(K) such that gxg−1 is a diagonal matrix; b) nilpotent (x = xn

) if xm = 0,that is for some positive integer m all eigenvalues of the matrix xm

are zero; c) unipotent (x = xu), if x−1n is nilpotent, i.e. x is the matrix whoseonly eigenvalues are 1’s.

Just like with the odd and even numbers the above matrices, if they exist,form closed disjoint subsets of GLn(K), e.g. all x, y ∈Mn(K) commute; if x, yare semisimple so is their sum and the product, etc. Most important for us isthe following fact:

Proposition 3.9. Let x ∈ GLn(K ). Then ∃ xu and xs such that x =xsxu = xuxs Both x sand xu are determined by the above conditions uniquely.

The proof can be found in Ref.[49, p.96]. This proposition is in fact acorollary of the Lie-Kolchin theorem which is of major importance for us. Toformulate this theorem we need to introduce yet another couple of definitions.In particular, if A and B are closed (finite) subgroups of the algebraic groupG one can construct the group (A,B) made of commutators xyx−1y−1, x ∈ A,y ∈ B . With help of such commutators the following definition can be made

Definition 3.10. The group G is solvable if its derived series terminates inthe unit element e. The derived series is being defined inductively by D(0)G =G,D(i+1)G = (D(i)G,D(i)G), i ≥ 0.

Such a definition implies that the algebraic group G is solvable if and only ifthere exists a chain G = G(0) ⊃ G(1) ⊃ ···G(n) = e for which (G(i), G(i)) ⊂ Gi+1

(0 ≤ i ≤ n), Ref.[49, p.111]. Finally,

23

Definition 3.11.The group is called nilpotent if E(n)G = e for some n, whereE(0) = G, E(i+1) = (G, E(i)G).

Such a group is represented by the nilpotent matrices. Based on this defini-tion, it is possible to prove that every nilpotent group is solvable [40, page 112].These results lead us to the Lie-Kolchin theorem of major importance

Theorem 3.12. (Lie and Kolchin, Ref.[49, p.113]) Let G be connectedsolvable algebraic group acting on a projective variety X. Then G has a fixedpoint in X.

In view of the Remark 3.8., we know that such G is a subgroup of GLn(K).Moreover, GLn(K) has at least another subgroup, called semisimple, for whichTheorem 3.12. does not hold. In this case we have the following

Definition 3.13. The group G is semisimple if it has no closed connectedcommutative normal subgroups other than e.

Such a group is represented by the semisimple, i.e. diagonal (or torus),matrices while the members of the unipotent group are represented by the uppertriangular matrices with all diagonal entries being equal to 1. In view of theTheorem 3.12, the unipotent group is also solvable and, accordingly, there mustbe an element B of such a group fixing the flag f0 defined by Eq.(3.11), i.e.Bf0 = f0. Let now g ∈ GLn(K ). Then, naturally, gf0 = f where f 6= f0.From here we obtain, f0 = g−1f . Next, we obtain as well, Bg−1f = g−1fand, finally, gBg−1f = f . Based on these results, it follows that gBg−1 = Bis also an element of GLn(K) fixing the flag f , etc. This means that allsuch elements are conjugate to each other and form the Borel subgroup. Weshall denote all elements of this sort by B. These are made of upper triangularmatrices belonging to GLn(K). Surely, such matrices satisfy Proposition 3.9.The quotient group G/B will act transitively on X . Since this quotient is alsoa linear algebraic group, it is as well a projective variety called the flag variety,Ref.[48, p.176]

Remark 3.14. The flag variety is directly connected with the Schubert vari-ety, Ref.[50, p.124].The Schubert varieties were considered earlier, in our work,Ref.[51], in connection with the exact combinatorial solution of the Kontsevich-Witten (K-W) model. Hence, the above remark naturally leads us to the com-binatorial approach to problems we are discussing in this part of our work andin Part IV. Additional details on connections with K-W model will becomeapparent in Section 4.

By now it should be clear that the group G is made out of at least twosubgroups: B, just described, and N . The maximal torus T subgroup of G canbe defined now as T = B ∩ N . This fact allows us to define the Weyl groupW = N/T . Although this group has the same name as that discussed in theAppendix, its true meaning in the present context requires some explanations.They will be provided below.

This is done in several steps. First, using results of Appendix we notice thatthe ”true” Weyl group is made of reflections, i.e. involutions of order 2. Fol-lowing Tits, Ref.[32], we introduce a quadruple (G,B,N, S) (the Tits system)where S is the subgroup of W made of elements such that S = S−1 and 1 /∈ S.

24

Such a subroup always exists for the compact Lie groups considered as sym-metric spaces. Then, it can be shown that G = BWB (Bruhat decomposition)and, moreover, that the Tits system is isomorphic to the Coxeter system, i.e.to the Coxeter reflection group. The full proof can be found in the monographby Bourbaki, Ref.[32], Chr.6, paragraph 2.4.

Second, since W = N/T , it is of interest to see the connection (if any)between W and the quotient G/B = BWB/B = [B (N/T )B] /B. In view ofthe fact that T = B ∩ N, suppose that N commutes with B. Then we wouldhave G/B ≃ (N/T )B and, since B fixes the flag f , we are left with the actionof N on the flag. In view of the rule M f(z) = f(Mz), and noticing thatthe diagonal matrix T (the centralizer) can be chosen as a reference (identity)transformation, we conclude that the commuting matrix N (the normalizer)should permute tai . Consider an application of this rule to the monomial ul =ul1

1 · · · ulnn ≡ zl1a1

1 · · · zln1ann . For such a map the character c(t) is given by

c(t) = t<l·a>, (3.12)

where, in accord with Eq.(3.5), < l · a >=∑

i liai with both li and ai beingsome integers. Following Ref.[52], let us consider the limit t → 0 in the aboveexpression. Clearly, we obtain:

c(t) =

1 if < l · a >= 00 if < l · a > 6= 0

. (3.13)

Evidently, the equation < l · a >= 0 describes a hyperplane or, better, a setof hyperplanes for a given vector a. In view of Eq.(3.5), such a set forms atleast one polyhedral cone (or chamber in the terminology of Appendix). Theseresults can be complicated a little bit by introducing a subset I ⊂ 1, ..., nsuch that, say, only those l′is which belong to this subset satisfy < l · a >= 0.Naturally, one obtains the one- to- one correspondence between such subsetsand earlier defined flags. Clearly, the set of such constructed monomials formsan invariant of the torus group action as discussed in in Part II. It remainsto demonstrate that the Weyl group W = N/T permutes ai’s thus forming anorbit transitively ”visiting” different hyperplanes. This will be demonstratedmomentarily. Before doing this, we would like to change the rules of the gameslightly13. To this purpose, we would like to replace the limiting t→ 0 procedureby the procedure requiring t→ ξ with ξ being the nontrivial n-th root of unity.After such a replacement we are entering the domain of the pseudo-reflectiongroups discussed in Part II. Thus, replacing t by ξ causes us to change the rule,Eq.(3.13), as follows:

c(ξ) =

1 if < l · a >= 0 modn0 if < l · a > 6= 0

. (3.14)

At this point it is appropriate to recall Eq.(I,3.11a). In view of this equation,we shall call the equation < l · a >= n as the Veneziano condition while the

13Such change of rules is consistent with arguments by Kirillov to be discussed in the nextsubsection.

25

Kac − Moody − Bloch − Bragg (K − M − B − B) condition, Eq.(I, 3.22), canbe written now as <l·a>= 0 modn .

The results of Appendix (part c)) indicate that the first option (the Venezianocondition) is characteristic for the standard Weyl-Coxeter (pseudo) reflectiongroups while the second is characteristic for the affine Weyl-Coxeter groupsthus leading to the Kac-Moody affine Lie algebras as discussed in Part II.

At this moment we are ready to demonstrate that W = N/T is indeed theWeyl reflection group. Even though the full proof can be found, for example, inthe monograph by Bourbaki, Ref.[32], still it is instructive to provide qualitativearguments exhibiting the essence of the proof (different from that given byBourbaki who use the Tits system).

Let us begin with an assembly of (d+ 1) × (d+ 1) matrices with complexcoefficients.They belong to the group GLd+1(C). Consider a subset of all diag-onal matrices and, having in mind physical applications, let us assume that thediagonal entries are made of n − th roots of unity ξ. Taking into account theresults on pseudo-reflection groups as discussed in Appendix (part d)) to PartII, each diagonal entry can be represented by ξk with 1 ≤ k ≤ n − 1 so thatthere are (n− 1)

d+1different diagonal matrices- all commuting with each other.

Among these commuting matrices we would like to single out those which haveall ξk ′s the same. Evidently, there are n− 1 of them. They are effectively theunit matrices and they are forming the centralizer of W . The rest belongs tothe normalizer.14 The number (n− 1)

d+1/(n−1) = (n− 1)

dwas obtained ear-

lier, e.g. see the discussion whoch follows Eq.(1.7) ( and replace 2m by n) andthe discussion which follows this equation. This is not just a mere coincidence.In the next section we shall provide some refinements of this result motivatedby physical considerations. It should be clear already that we are discussingonly the simplest possibility (of cubic symmetry) for the sake of illustration ofgeneral principles. Clearly, the zonotope construction, introduced earlier allowsus to transfer our reasonings to more general cases.

Next, let us consider just one of the diagonal matrices T whose entries areall different and are made of powers of ξ. Let g ∈ GLd+1(C) and consideran automorphism: F(T ) := gTg−1. Along with it, we would like to consideran orbit O(T ) := gTC where C is any of the diagonal matrices belonging toearlier discussed centralizer.15 Clearly, O(T ) = gTg−1gC = F(T )gC = F(T )C.Denote now T =T1 and consider another matrix T2 belonging to the same setand suppose that there is such matrix g12 that T2C = F(T )C. If such a matrixexist, it should belong to the normalizer and, naturally, the same arguments canbe used to T3, etc. Hence, the following conclusions can be drawn. If we hadstarted with some element T1 of the maximal torus, the orbit of this elementwill return back and intersect the maximal torus in finite number of points (in

14As with Eq.(3.12), one can complicate matters by considering matrices which have severaldiagonal entries which are the same. Then, as before, one should consider the flag system wherein each subsystem the entries are all different. The arguments applied to such subsystemswill proceed the same way as in the main text.

15Presence of C factor underscores the fact that we are considering the orbit of the factor-group W = N/T .

26

our case the number of points is exactly (n− 1)d). By analogy with the theory of

dynamical systems, we can consider these intersection points of the orbit O(T )with the T -plane as the Poincare′ crossections. Hence, as it is done in the caseof dynamical systems (e.g. see Section 2.2) we have to study the transition mapbetween these crossections. The orbit associated with such a map is preciselythe orbit of the Weyl groupW . It acts on these points transitively [49, p.147], asrequired. Provided that the set of fixed point exists, such arguments justify thedynamical interpretation of the Weyl’s character formula presented in Section2.2. The fact that such fixed point set does exist is guaranteed by the Theorem10.6. by Borel, Ref.[47]. Its proof relies heavily on the Lie-Kolchin theorem (ourTheorem 3.12.).

3.2 Coadjoint orbits

Thus far we were working with the Lie groups. To move forward we need touse the Lie algebras associated with these groups. In what follows, we expectour readers familiarity with basic relevant facts about the Lie groups and Liealgebras which can be found in the books by Serre [53], Humphreys [54] andKac [36]. First, we notice that the Lie algebra matrices hi associated with theLie group maximal tori Ti (that is with all diagonal matrices considered earlier)are commuting with each other thus forming the Cartan subalgebra, i.e.

[hi, hj ] = 0. (3.15)

The matrices belonging to the normalizer are made of two types xi and yi

corresponding to the root systems ∆+ and ∆− defined in Appendix . The fixedpoint analysis described at the end of previous section is translated into thefollowing set of commutators

[xi, yj ] =

hi if i = j0 if i 6= j

, (3.16a)

[hi, xj ] =< α∨i , αj > xj , (3.16b)

[hi, yj ] = − < α∨i , αj > yj , (3.16c)

i = 1, ..., n. To insure that the matrices (operators) x′is and y′is are nilpotent(that is their Lie group ancestors belong to the Borel subgroup B) one mustimpose two additional constraints. According to Serre [53] these are:

(adxi)−<α∨

i ,αj>+1(xj) = 0, i 6= j (3.16d)

(adyi)−<α∨

i ,αj>+1(yj) = 0, i 6= j. (3.16e)

where adX Y = [X,Y ]. From the book by Kac [36] one finds that exactlythe same relations characterize the Kac-Moody affine Lie algebra. This factis in accord with general results presented earlier in this work and in Part IIand is of major importance for develpment of our formalism. In particular,for the purposes of this development it is important to realize that for each

27

i Eq.s(3.16a-c) can be brought to form (upon rescaling) coinciding with theLie algebra sl2(C)16 and, if we replace C with any closed number field F,then all semisimple Lie algebras are made of copies of sl2(F) [54, p.25].The Liealgebra sl2(C) is isomorphic to the algebra of operators acting on differentialforms living on the Hodge-type complex manifolds [55]. This observation wasabsolutely essential for development of physical applications in Part II.

Connections with Hodge theory can be also established through the methodof coadjoint orbits. We would like to discuss this method now. We beginby considering an orbit in the Lie group. It is given by the Ad operator,i.e. O(X) =AdgX = gXg−1 where g ∈ G and X ∈g with G being theLie group and g its Lie algebra. For compact groups globally and for non-compact locally every group element g can be represented via the exponen-tial, e.g. g(t)=exp(tXg) with Xg ∈g. Accordingly, for the orbit we can writeO(X) ≡ X(t) = exp(tXg)X exp(−tXg). Since the Lie group is a manifold M,the Lie algebra forms the tangent bundle of the vector fields at given point ofM. In particular, the tangent vector to the orbit X(t) is determined, as usual,by TO(X) = d

dtX(t)t=0 = [Xg, X ] = adXg

X. Now we have to take into accountthat, actually, our orbit is made for a vector X coming from the torus, i.e.T = exp(tX). This means that when we consider the commutator [Xg, X ] itwill be zero for Xgi

= hi and nonzero otherwise. Consider next the Killing formκ(x, y) for two elements x and y of the Lie algebra:

κ(x, y) = tr(adx ady). (3.17)

From this definition it follows that

κ([x, y], z) = κ(x, [y, z]). (3.18)

The roots of the Weyl group can be rewritten in terms of the Killing form [54].Its purpose is to define the scalar multiplication between vectors belonging to theLie algebra and, as such, it allows one to determine the notion of orthogonalitybetween these vectors. In particular, if we choose x→ X and y, z ∈ hi, then it isclear that the vector tangential to the orbitO(X) is going to be orthogonal to thesubspace spanned by the Cartan subalgebra. This result can be reinterpretedfrom the point of view of symplectic geometry due to work of Kirillov [57].Tothis purpose we would like to rewrite Eq.(3.18) in the equivalent form, i.e.

κ(x, [y, z]) = κ(x, adyz) = κ(ad∗xy, z) (3.19)

where in the case of compact Lie group ad∗xy actually coincides with adxy . The

reason for introducing the asterisk * lies in the following chain of arguments.In Eq.(A.1) of Appendix (and in Eq.(3.5)) we introduced vectors from the dualspace. Such a construction is possible as long as the scalar multiplication isdefined. Hence, for the orbit AdgX there must be a vector ξ in the dual spaceg∗ such that equation

κ(ξ,AdgX) = κ(Ad∗gξ,X) (3.20)

16This fact is known as Jacobson-Morozov theorem [56].

28

defines the coadjoint orbit O∗(ξ)=Ad∗gξ. Accordingly, for such an orbit there

is also the tangent vector TO∗(ξ)=ad∗gξ to the orbit and, clearly, we have

κ(ξ, adXgX) = κ(ad∗

gξ,X). In the case if we are dealing with the flag space, the

family of coadjoint orbits will represent the flag space structure as well. Next, letx ∈g

∗ and ξ1, ξ2 ∈ TO∗(x), then consider the properties of the (symplectic) formωx(ξ1, ξ2) to be determined explicitly momentarily. For this purpose we need tointroduce some notations, e.g. ad∗

gx = f(x, g), etc. so that for g1 and g2 ∈g one

has ξi = f(x, gi), i = 1, 2. Then, one can claim that for the compact Lie groupand the associated with it Lie algebra ωx(ξ1, ξ2) = κ(x, [g1, g2]). Indeed, usingEq.(3.18) we obtain: κ(x, [g1, g2]) = κ(ξ1, g2) = −κ(x, [g2, g1]) = −κ(ξ2, g1).Thus constructed form defines the symplectic structure on the coadjoint orbitO∗(x) since it is closed, skew -symmetric, nondegenerate and is effectively in-dependent of the choice of g1 and g2.The proofs can be found in the literature[58].The obtained symplectic manifold Mx is the quotient g/gh with gh beingmade of vectors of theCartan subalgebra so that for such vectors, by construc-tion, ωx(ξ1, ξ2) = 0. From the point of view of symplectic geometry, the pointsfor which ωx(ξ1, ξ2) = 0 correspond to the critical points for the velocity vectorfield on the manifold Mx. I.e. these are the points at which the velocity fieldvanishes. They are in one-to one correspondence with the fixed points of theorbit O(X). This fact allows us to use the Poincare′-Hopf index theorem (ear-lier used in our works on dynamics of 2+1 gravity [59]) in order to obtain theEuler characteristic χ for such manifold as the sum of indices of vector fieldsexisting on Mx.We shall provide more details related to this observation belowin Section 3.4.

To complete the above discussion, following Atiyah [60], we notice that everynonsigular algebraic variety in projective space is symplectic. The symplectic(Kahler) structure is inherited from that in the projective space. The complexKahler structure for the symplectic (Kirillov) manifold is actually of the Hodge-type. This comes from the following observations. First, since we have used theKilling form to determine the Kirillov symplectic form ωx and since the sameKilling form is used for the Weyl reflection groups [58], the induced unitaryone dimensional representation of the torus subgroup of GLn(C) is obtainedaccording to Kirillov [57] by simply replacing t by the root of unity in Eq.(3.12).This is permissible if and only if the integral of two-form

∫

γωx taken over any

two dimensional cycle γ on the coadjoint orbit O∗(x) is an integer. But thisis exactly the condition which makes the Kahler complex structure that of theHodge type [55].

3.3 Construction of the moment map using methods of

linear programming

In this subsection we are not employing the definition of the moment mappingused in symplectic geometry [61]17. Instead, we shall rely heavily on works byAtiyah [60,62] with only slightest refinement coming from noticed connections

17Evidently, we are using the same thing anyway.

29

with the linear programming not mentioned in his papers and in literature onsymplectic geometry. In our opinion, such a connection is helpful for betterphysical understanding of mathematical methods discussed in this paper whichmight be useful for applications in other disciplines.

Using Definition 1.1. of Section 1. we call the subset of Rn a polyhedronP if there exist m× n matrix A (with m < n) and a vector b ∈ Rm such thataccording to Eq.(1.8) we have

P = x ∈ Rn | Ax ≤ b.

Since each component of the inequalities Ax ≤ b determines the half spacewhile the equality Ax = b -the underlying hyperplane, the polyhedron is anintersection of finitely many halfspaces. The problem of linear programmingcan be formulated as follows [63] : for the linear functional H[x]=cT ·x find maxH[x] on P provided that the vector c is assigned. It should be noted that thisproblem is just one of many related problems. It was selected only because ofits immediate relevance. Its relevance comes from the fact that the extremumof H[x] is achieved at least at one of the vertices of P . The proof of this we omitsince it can be found in any standard textbook on linear programming, e.g. see[64] and references therein. This result does not require the polyhedron to becentrally symmetric. Only convexity of the polyhedron is of importance. This isphysically plausible since, for instance, reflexive polyhedra discussed in Section1 in connection with mirror symmetry do not require such central symmetryas can be seen from two dimensional examples presented in Ref.[65, p.100].

To connect this optimization problem with results of our paper we constrainx variables to integers, i.e. to Zn. Such a restriction is known in literature asinteger linear programming. In our case, it is equivalent to considering sym-plectic manifolds of Hodge-type (e.g. read page 11 of Atiyah’s paper, Ref.[60]).Hence, existence of mirror symmetry as well as the method of coadjoint orbitsboth require the underlying symplectic manifolds to be of Hodge-type. Thishas a deep physical reason which will become clear when we shall discuss theKhovanskii-Pukhlikov correspondence in the next section.

As a warm up exercise, following Fulton, Ref.[40], let us consider a defor-mation retract of complex projective space CPn which is the simplest possibletoric variety [40,41]18. Such a retraction is achieved by using the map :

τ : CPn → Pn≥ = Rn+1

≥ \ 0/R+

explicitly given by

τ : (z0, ..., zn) 7→ 1∑

i |zi|(|z0| , ..., |zn|) = (t0, ..., tn) , ti ≥ 0. (3.21)

The map τ by design is onto the standard n-simplex : ti ≥ 0, t0+ ...+tn = 1. Tobring physics to this discussion, let us consider the Hamiltonian for the harmonic

18Although such a construction was introduced in Part II, we write it down explicitly anywayfor the sake of uninterrupted reading.

30

oscillator. In the appropriate system of units we can write it as H =m(p2 + q2).More generally, for finite set of oscillators, i.e. for the ”truncated” bosonic string,we have: H[z] =

∑

i mi |zi|2 , where, following Atiyah [60], we introduced thecomplex zj variables via zj = pj + iqj. Let now such a Hamiltonian system (thetruncated string) possess the finite fixed energy E . Then we obtain:

H[z] =∑n

i=0mi |zi|2 = E . (3.22)

It is not difficult to realize that the above equation actually represents the CPn

since the points zj can be identified with the points eiθzj in Eq.(3.22) whilekeeping the above expression form- invariant. In such a case one is saying thatthe reduced phase space for this model is CPn as discussed in Section 7.6. ofPart II.We can map such a model of CPn back into the simplex using the mapτ . Since CPn is the simplest toric variety [40,41], if we let zj to ”live” in sucha variety it will be affected by the torus action as discussed earlier in thissection. This means that, in general, the masses in Eq.(3.22) may change and,accordingly, the energy. Only if we constrain the torus action to the simplex(or, more generally, to the polyhedron as described by Fulton, Ref.[40],) willthe energy be conserved. Evidently, such a constraint is compatible with theoriginal idea of identification of points zj with eiθzj . The fixed points of suchdefined torus action are roots of unity according to Eq.(3.10). In general, theexistence of at least one fixed point is guaranteed for the linear algebraic groupby the Theorem 10.6, Ref.[47]. With such defined torus action, |zi|2 is just somepositive number, say, xi. The essence of the moment map lies exactly in suchidentification19. Hence, we obtain the following image of the moment map:

H[x] =∑n

i=0mixi, (3.23)

where we have removed the energy constraint for a moment thus making H[x]to coincide with earlier defined linear functional to be optimized. Now wehave to find a convex polyhedron on which such a functional is going to beoptimized. Thanks to works by Atiyah [60,62] and Guillemin and Sternberg [66],this task is completed already. Naturally, the vertices of such a polyhedron arethe critical points of the moment map. Then, the theorem of linear programingstated earlier guarantees that H[x] achieves its maximum at least at some ofits vertices. Delzant [67] had demonstrated that this is the case without use oflinear programming language.

It is helpful to illustrate the essence of above arguments by employing simplebut important example originally discussed by Frankel [68]. Consider a twosphere S2 of unit radius, i.e. x2 +y2 +z2 = 1, and parametrize this sphere usingcoordinates x =

√1 − z2 cosφ, y =

√1 − z2 sinφ , z = z. In section 4 we shall

demonstrate that the Hamiltonian for the free particle ”living” on such a sphereis given by H[z]=m (1 − z) so that equations of motion produce the circles oflatitude. These circles become (critical) points of equilibria at the north and

19More accurate definition is given in Section 4.

31

south pole of the sphere, i.e. for z = ±1. Evidently, these are the fixed points ofthe torus action. Under such circumstances our polyhedron is the segment [-1,1]and its vertices are located at ±1 (to be compared with discussion in Section1.). The image of the moment map H[x]=m (1 − x) acquires its maximum atx = 1 and the value x = 1 corresponds to two polyhedral vertices located at 1and -1 respectively. This doubling feature was noticed and discussed in detailby Delzant [67] whose work contains all needed proofs. These can be consideredas elaborations on much earlier results by Frankel [68]. The circles on the sphereare represening the torus action (e.g. read the discussion following Eq.(3.22))so that dimension of the circle is half of that of the sphere. This happens tobe a general trend : the dimension of the Cartan subalgebra (more accurately,the normalizer of the maximal torus) is half of the dimension of the symplecticmanifold M [39,67]. Incidentally, in the next subsection we shall see that theintegral of the Kirillov-Kostant symplectic two- form ωx over S2 is equal to 2 sothat the complex structure on the sphere is that of the Hodge type as required.Also, the symplectic two- form ωx = 0 at two critical points. Generalization ofthis example to the multiparticle case will be discussed below and in Section 4.

The results discussed thus far although establish connection between thesingularities of symplectic manifolds and polyhedra do not allow us to discussthe fine details distinguishing between different polyhedra. Fortunately, this hasbeen to a large degree accomplished in Refs.[58,69]. Such a task is equivalent toclassification of all finite dimensional exactly integrable systems in accord withthe Lie groups and Lie algebras associated with them..

3.4 Calculation of the Euler characteristic

Using results just presented we are ready to calculate the Euler characteristicof the projective algebraic variety following ideas by Hopf [45] and Hopf andSamelson [46]. To begin, we notice that in the case of vector fields on S2

discussed in the previous subsection there are two fixed points. The Poincare′-Hopf fixed point theorem (extensively used in our earlier work on 2+1 gravity,Ref.[59]) tells us that χ is the sum of indices of the vector (or line) fields foliatingmanifold M. In our case, the index of each critical point is known to be 1so that χ = 2 as required20. In the case of S2 the Darboux coordinates canbe chosen as φ and z with 0 ≤ φ < 2π and z ∈ [-1,1]. The volume formΩ is dφ ∧ dz so that the phase space is effectively the product R × S1. Wewould like to construct now a dynamical system whose Darboux coordinatesare t1, ..., tn;φ1, ..., φn. If in the case of S2 the z coordinate varied in thesegment [-1,1], now we shall assume that the point t =t1, ..., tk can varyinside some polytope P ⊂ Rk including its boundaries. For our purposes, inview of Eq.(3.22), it is sufficient to consider only some simplex ∆k living inRk. This happens when all masses in Eq.(3.22) are the same so that using

20Incidentally, if following Delzant, Ref.[67], we divide the number of polyhedral verticesby factor of 2, then using Eq.(1.7) with 2m replaced by 1 we shall reobtain the result χ = 2.More formally, we can say that the cardinality |G|= 1

2dimM. That this is indeed the case in

general was provern by Delzant.

32

Eq.s(3.21) and (3.22) we obtain equation for the simplex. In the case of S2

we can think of z coordinate as deformation retract for S2. One can say thatS2 is the inflated symplectic manifold corresponding to the segment [-1,1], i.e.S2 ∼ R × S1. Accordingly, we can say that CPk ∼ ∆k × S1 × · · ·S1. The

Darboux coordinates t1, ..., tk;φ1, ..., φk → t12

1 eiφ1 , ..., t

12

k eiφk ,

√

1 −∑i ti ≡z1, ..., zk, zk+1, provided that t1+...+tk+1 = 1.These results are in accord withEq.(3.21). Such a choice of coordinates realizes CPk as the space of equivalenceclasses

|z1|2 + · · · + |zk|2 + |zk+1|2 = 1 , zi ∼ eiφizi , i = 1 − k. (3.24)

of the points lying on the sphere S2k+1 in Ck+1 (we used this realization ofCPk already in Part II). In accord with previous subsection, it is the reducedphase space (the reduced symplectic manifold Mred) for our dynamical system.

Following Section 1., it is of interest to consider the inflated simplex n∆k

living on the lattice Zk. Accordingly, we can consider the associated with itthe inflated symplectic manifold M. The indices of critical points of such amanifold produce its Euler characteristic χ. Irrespective to locations of criticalpoints on such a manifold, the point t should have coordinates such that t1 +... + tk = n. If P is the rational polytope these coordinates should be someintegers. Accordingly, one has to count the number of solutions to the equationt1 + ...+ tk = n in order to determine the number p(k, n) of such critical points.This number we know already since it is given by Eq.(1.2). Accordingly, forphysically interesting case associated with our interpretation of the Venezianoamplitudes we obtain, p(k, n) = χ. These rather simple arguments are useful tocompare with extremely sophisticated proofs of the same result using methodsof algebraic geometry, e.g. see Refs.[40-42]. These methods are of importancehowever in case if one is interested in computation of some observables as itis done earlier, for example, for the Witten-Kontsevich model [51]. More onthis will be said below and in Part IV. Obtained results provide us with toolsneeded for symplectic treatment of the Veneziano amplitudes and for restorationof generating function associated with these amplitudes. This is accomplishedin the next section.

4 Exact solution of the Veneziano model :sym-

plectic treatment

4.1 The Moment map, the Duistermaat-Heckman formula

and the Khovanskii-Pukhlikov correspondence

4.1.1 General remarks

We have mentioned already number of times mathematical connections betweenthe Veneziano amplitudes (and the associated with them Veneziano partitionfunction) and dynamical systems. We would like to summarize these results

33

now. First, already in Part I we emphasized that the development in this se-ries of work is motivated in part by two major obseravations. These are : a)the unsymmetrized Veneziano amplitude can be looked upon as the Laplacetransform of the partition function obtained by quantization of finite set of har-monic oscillators as described in the work by Vergne [3], b) the unsymmetrizedVeneziano amplitude can be interpreted as one of the periods associated withhomology cycles on the variety of Fermat-type. These observations are sufficientfor development of both symplectic and supersymmetric approaches leading torestoration of the underlying physical model producing the Veneziano-like am-plitudes. In Part II we strengthened these observations by invoking theorems bySolomon and Ginzburg (Theorems 2.2. and 2.5. respectively). Also, in Part IIusing results by Shepard and Todd and Serre we provided enough eveidence forthe Veneziano partition function to be supersymmetric.Using these results weobtained exact solution for the Veneziano model, i.e. we have obtained the par-tition/generating function for this model whose observables are unsymmetrizedVeneziano amplitudes. In this work we provided additional details directing ustowards alternative (symplectic) interpretation of this partition function. Theseinclude: a) zeta function by Ruelle, b) method of coadjoint orbits and c) themoment map. Connections between supersymmetric and symplectic descrip-tions can be deduced using well written monograph by Berline, Getzler andVergne [70]. In view of this, to avoid excessive size of our paper, it is sufficientto emphasize only things of immediate relevance. In particular, we would liketo discuss now the Duistermaat-Heckman formula.

4.1.2 The Duistermaat-Heckman formula

Although the description of the Duistermaat-Heckman (D-H) formula can befound in many places, we would like to discuss it now in connection with earlierobtained results. To this purpose, using Subsection 3.4. let us consider onceagain the simplest dynamical model discussed there. The volume form Ω forthis model is given by Ω :=dθ ∧ dz so that

∫

S2 Ω = 4π as expected. With helpof this form the equation for the moment map can be obtained. According tothe standard rules [39,61], given that ξ = ∂

∂θ, we obtain,

i(ξ)Ω = dz. (4.1)

The Hamiltonian H, i.e. the moment map, is given in this case by H=z. Con-sider now the integral I(β) of the type

I(β) =

∫

Mred

Ω exp(−βH) =1

β(exp(β) − exp(−β)). (4.2)

In this integral the reduced phase space is Mred = S2/S1 so that Ω = dz and,as before, z ∈ [−1, 1]. Eq.(4.2) is essentially the D-H formula! We would liketo explain this fact in some detail. In view of the results of Subsection 3.3. we

34

know that the moment map H achieves its extrema at the vertices of P . Sincein our case P is the segment [−1, 1], indeed, H achieves its extrema at both 1and -1 so that the r.h.s. of Eq.(4.2) is in fact the sum over the vertices of Ptaken with the appropriate weights. The D-H formula provides exactly the sameanswer. Indeed let M ≡M2n be a compact symplectic manifold equipped withthe momentum map Φ : M → R and the (Liouville) volume form dV =

(

12π

)n 1n!

Ωn. According to the Darboux theorem, the two-form Ω can be presented locallyas: Ω =

∑nl=1 dql ∧dpl . Suppose that such a manifold has isolated fixed points

p belonging to the fixed point set V associated with the isotropy subgroup G(Definition 3.5.) acting on M . Then, in its most general form, the D-H formulacan be written as [39,61]

∫

M

dV eΦ =∑

p∈V

eΦ(p)

∏

j aj,p

(4.3)

where a1,p, ..., an,p are the weights of the linearized action of G on TpM . Usingthe Morse theory, Atiyah [62] and others [61] have demonstrated that it issufficient to keep terms up to quadratic in the expansion of Φ around givenp. In such a case the moment map looks exactly like that given in Eq.(3.22).Moreover, the coefficients a1,p, ..., an,p are just ”masses”mi in Eq.(3.22). We putquotation marks around masses since they can be both positive and negative.With these remarks, it should be obvious that Eq.(4.2) is the D-H formula. Itshould be noted that although in Eq.(4.3) the space M is not reduced, Eq.(4.2)can be written without requirement of reduction as well. For this it is sufficientto consider in Eq.(4.2) the form Ω = 1

2πdθ ∧ dz. Hence, indeed, Eq.(4.2) is the

D-H formula. Consider now the limiting case β → 0+ of Eq.(4.2). Then, weobtain

I(β → 0+) = 2. (4.4)

But 2 is the Euclidean volume of the polytope P , in our case, the length of thesegment [−1, 1]. This is in accord with general result obtained by Atiyah [60].Now we would like to generalize this apparently trivial result in several direc-tions. First, we would like to blow up the sphere so that its diameter would be2m. Second, we would like to consider a collection of such spheres with respec-tive diameters 2mi, i = 1−d. For such a collection we can consider 2 situations: a) the total energy E for the Hamiltonian H=

∑

i zi is not conserved and b)the total energy is conserved, e.g. see Eq.(3.22). The first case is nonphysicalbut, apparently, is relevant for theory of singularities of differentiable maps andis related to the computation of the Milnor number. This issue was discussedin our earlier work, Ref.[71]. The second case is physically relevant. Hence, wewould like to discuss it in some detail. Both cases can be found as exercises onpage 50 in the book by Guillemin, Ref.[38]. In discussing the second case bothGuillemin, Ref.[38], and Audin, Ref.[61 ], notice that the action for the torusT d = S1 × · · · × S1 on such Hamiltonian system is diagonal (e.g. Section 3)and is made of d-tuples (eiθ1 , ..., eiθd) subject to the constraint eiθ1 · · · eiθd = 1.This constraint is actually the Veneziano condition discussed in Section 3, e.g.see Eq.(3.14).

35

Based on the information just mentioned, we would like to be more specificnow. To this purpose, following Vergne [72] and Brion [6] we would like toconsider the simplest nontrivial case of the integral of the form

I(k) =

∫

k∆

dx1dx2 exp−(y1x1 + y2x2) (4.5)

where k∆ is dilated standard simplex with coefficient of dilation k. Followingthese authors, calculation of this integral can be done exactly with the result,

I(k) =1

y1y2+

e−ky1

y1(y1 − y2)+

e−ky2

y2(y2 − y1). (4.6)

As in earlier case of Eq.(4.2), the obtained result fits the D-H formula, Eq.(4.3),and, as before, in the limit: y1, y2 → 0 some calculation produces the anticipatedresult : V olk∆ = k2/2! , in accord with Eq.(4.4). In view of results of PartsI, II and this work, this integral is of relevance to calculation of Venezianoamplitudes (and it does have symplectic meanning !): The standard simplex ∆in the present case is just the deformation retract for the Fermat (hyper)surfaceassociated with calculation of the Veneziano (or Veneziano-like) amplitudes. Therelevance of this integral to the Veneziano amplitude is far from superficial aswe would like to discuss now.

4.1.3 The Khovanskii-Pukhlikov correspondence and calculation ofχ

The Khovanskii-Pukhlikov correspondence can be understood based on the fol-lowing generic example. Following Ref.[73] we would like to compare the integral

I(z) =

t∫

s

dxezx =etz

z− esz

zwith the sum S(z) =

t∑

k=s

ekz =etz

1 − e−z+

esz

1 − ez,

(4.7)where Eq.(1.4) was used for calculation of S(z).One can pose a problem : is thereway to transform the integral I into the sum S ? Clearly, we are interested insuch a transform in view of the fact that the Ehrhart polynomial computes thenumber of lattice points of the dilated polytope while the D-H integral can beused only for calculation of the Euclidean volumes of such polytopes as we justdemonstrated on simple examples. The positive answer to the above questionwas found by Khovanskii and Pukhlikov [74] and refined by many others, e.g.see Ref.[73]. Before discussing their work, we would like to write down thediscrete analog of the result, Eq.(4.6). It is given by

S(k) =1

1 − e−y1

1

1 − e−y2+

1

1 − ey1

e−ky1

1 − ey1−y2+

1

1 − ey2

e−ky2

1 − ey2−y1(4.8)

=∑

(l1,l2)∈k∆

exp−(y1l1 + y2l2).

36

This result can be obtained rather straightforwardly using Brion’s formula forthe generating function for polytopes. It is given by earlier discussed Eq.(1.11)and, hence, it is in complete accord with this more general equation. Somecomputational details can be found in the monograph by Barvinok, Ref.[7].Following his exposition we would like to discuss some physics behind theseformal calculations. For this we need to use the definition of the monoid Sσ,Eq.(3.3), introduced earlier. In view of the Remark 9.9.(Part II) the set a1, ..., ak

forms a basis of the vector space V so that the monoid Sσ defines a rationalpolyhedral cone σ. Thanks to the theorem by Brion [6,7] the generating functionin the l.h.s. of Eq.(1.11) can be conveniently rewritten as

f(P ,x) =∑

m∈P∩Zd

xm =∑

σ∈V ertP

xσ (4.9a)

so that for the dilated polytope it reads as follows

f(kP ,x) =∑

m∈kP∩Zd

xm =n∑

i=1

xkvi

∑

σi

xσi . (4.9b)

In the last formula the summation is taking place over all vertices whose locationis given by the vectors from the set v1, ...,vn. This means that in actualcalculations one can first calculate the contributions coming from the cones σi

of the undilated (original) polytope P and only then one can use this equationin order to get the result for the dilated polytope. Let us apply these generalrules to our specific problem of computation of S(k) in Eq.(4.8). We have oursimplex with vertices in x-y plane given by the vector set v1=(0, 0), v2=(1,0),v3=(0,1) where we have written the x coordinate first. For this case we have 3cones: σ1 = l2v2 + l3v3 ; σ2 = v2 + l1(−v2)+ l2(v3−v2);σ3 =v3 + l3(v2−v3)+l1(−v3);l1, l2 , l3 ∈ Z+ . In writing these expressions for the cones wehave taken into account that, according to Brion, when making calculationsthe apex of each cone should be chosen as the origin of the coordinate system.Calculation of contributions to generating function coming from σ1 is the moststraightforward. Indeed, in this case we have x = x1x2 = e−y1e−y2 . Now, thesymbol xσ in Eq.s(4.9) should be understood as follows. Since σi , i = 1 − 3,is actually a vector, it has components, like those for v1,etc. We shall write

therefore xσ = xσ(1)1 · · · xσ(d)

d where σ(i) is the i-th component of such a vector.Under these conditions calculation of the contributions from the first cone withthe apex located at (0,0) is completely straightforward

∑

(l2,l3)∈Z2+

xl21 x

l32 =

1

1 − e−y1

1

1 − e−y2(4.10)

since it is reduced to the computation of the infinite geometric progressions. Butphysically, the above result can be looked upon as a product of two partitionfunctions for two harmonic oscillators whose ground state energy was discarded.By doing the rest of calculations in the way just described we reobtain S(k) from

37

Eq.(4.8) as required. This time, however, we know that the obtained result isassociated with the assembly of harmonic oscillators of frequencies ±y1, ±y2 and±(y1 − y2) whose ground state energy is properly adjusted. The ”frequencies”(or masses) of these oscillators are coming from the Morse-theoretic consider-ations for the moment maps associated with the critical points of symplecticmanifolds as explained in the paper by Atiyah [62]. These masses enter intothe ”classical” D-H formula. It is just a classical partition function for a systemof such described harmonic oscillators living in phase space containing singu-larities. The D-H classical partition function, Eq.(4.6), has its quantum analog,Eq.(4.8), just described. The ground state for such a quantum system is degen-erate with degeneracy being described by the Kostant multiplicity formula. Tocalculate this degeneracy would require us to study the limiting case: y1, y2 → 0of Eq.(4.8) for S(k). Surprisingly, unlike the continuum case sudied in the previ-ous subsection, calculation of number of points belonging to the dilated simplexk∆ (which is equivalent to the calculation of the Kostant multiplicity formulaor,which is the same, to the computation of the Ehrhart polynomial or to theEuler characteristic χ of the associated projective toric variety) is very nontriv-ial in the present case. It is facilitated by the observation that in the limit s→ 0the following expansion can be used

1

1 − e−s=

1

s+

1

2+

s

12+O(s2). (4.11)

Rather lengthy calculation involving this expansion produces in the end theanticipated result for the Ehrhart polynomial:

∣

∣k∆ ∩ Z2∣

∣ = P (k, 2) =k2

2+

3

2k + 1. (4.12)

Obtained results and their interpretations are in formal accord with those byVergne, Ref.[3]. In her work no details (like those presented above) or physicalapplications are given however. At the same time, the results obtained thus farapparently are not in agreement with earlier obtained major result, Eq.(1.1).Fortunately, the situation can be corrected with help of the following theoremby Barvinok, Ref.[75].

Theorem 4.1. For the fixed lattice of dimensionality d there exist a poly-nomial time algorithms which, for any given rational polytope P , calculate thegenerating function f(P , x) with the result

f(P , x) =∑

m∈P∩Zd

xm =∑

i∈I

ǫixpi

(1 − xai 1) · · · (1 − xai d)(4.13)

where ǫ ∈ −1, 1, pi, ai j ∈ Zd and ai j 6= 0 ∀ i, j . In fact, ∀ i , the set ai 1 , ...,ai d forms a basis of Zd and I is the set 1, ..., nlabeling the vertices of P .

Remark 4.2. It is easy to check this result using Eq.(1.1) for n (or d) equalto 2 and comparing it with S(k) from Eq.(4.8). It should be noted however, that

38

Eq.(1.1) was obtained using some kind of combinatorial and supersymmetric ar-guments as explained in Part II while Eq.(4.8) is obtained exclusively based onuse of the bosonic formalism. It should be clear that both approaches leadingto the design of new model reproducing the Veneziano and Veneziano-like am-plitudes can be used in principle since they are essentially equivalent in view ofthe earlier mentioned Ref.[70].

At this point, finally, we are ready do discuss the Khovanskii-Pukhlikov cor-respondence. It should be considered as alternative to the method of coadjointorbits discussed in Section 3.2. Naturally, both methods are in agreement witheach other with respect to final results. Following Refs[38,72,73] we introducethe Todd operator (transform) via

Td(z) =z

1 − e−z. (4.14)

In view of Eq.(4.7), it can be demonstrated [73] that

Td(∂

∂h1)Td(

∂

∂h2)(

∫ t+h2

s−h1

ezxdx) |h1=h1=0=

t∑

k=s

ekz. (4.15)

This result can be now broadly generalized following ideas of Khovanskii andPukhlikov, Ref. [66?]. In particular, the relation

Td(∂

∂z) exp

(

n∑

i=1

pizi

)

= Td(p1, ..., pn) exp

(

n∑

i=1

pizi

)

(4.16)

happens to be the most useful. Applying it to

i(x1, ..., xk; ξ1, ..., ξk) =1

ξ1...ξk

exp(

k∑

i=1

xiξi) (4.17)

we obtain,

s(x1, ..., xk; ξ1, ..., ξk) =1

k∏

i=1

(1 − exp(−ξi))

exp(

k∑

i=1

xiξi). (4.18)

This result should be compared now with the individual terms on the r.h.s. ofEq.(1.11) on one hand and with the individual terms on the r.h.s of Eq.(4.3)on another. Evidently, with help of the Todd transform the exact ”classical”results for the D-H integral are transformed into the ”quantum” Weyl characterformula.

We would like to illustrate these general observations by comparing the D-Hresult, Eq.(4.6), with the Weyl character formula (e.g.see Eq.(1.11)), Eq.(4.8).To this purpose we need to use already known data for the cones σi , i = 1− 3,and the convention for the symbol xσ. In particular, for the first cone we

39

have already : xσ1 = xl11 x

l22 = [exp(l1y1)] · [exp(l2y2)]

21. Now we assemblethe contribution from the first vertex using Eq.(4.6). We obtain, [exp(l1y1)] ·[exp(l2y2)] /y1y2. Using the Todd transform we obtain as well,

Td(∂

∂l1)Td(

∂

∂l2)

1

y1y2[exp(l1y1)] · [exp(l2y2)] |l1=l2=0=

1

1 − e−y1

1

1 − e−y2.

(4.19)Analogously, for the second cone we obtain: xσ

2 = e−ky1e−l1y1e−l2(y1−y2) sothat use of the Todd transform produces

Td(∂

∂l1)Td(

∂

∂l2)

1

y1 (y1 − y2)e−ky1e−l1y1e−l2(y1−y2) |l1=l2=0=

1

1 − ey1

e−ky1

1 − ey1−y2,

(4.20)etc.

In Section 3.4. we sketched ideas behind calculations of Euler characteristicχ. It is instructive in the light of just obtained results to reobtain χ.

To accomplish the task is actually not difficult since it is based on the infor-mation we have presented already. Indeed, by looking at the last two equationsit makes sense to rewrite formally the partition function, Eq.(4.5), in the fol-lowing symbolic form

I(k, f) =

∫

k∆

dx exp (−f · x) (4.21)

valid for any finite dimension d. Since we have performed all calculations ex-plicitly for two dimensional case, for the sake of space, we only provide the ideabehind such type of calculation for any d 22. In particular, using Eq.(4.3) wecan rewrite this integral formally as follows

∫

k∆

dx exp (−f · x) =∑

p

exp(−f · x(p))d∏

i

hpi (f)

. (4.22)

Applying the Todd operator (transform) to both sides of this formal expressionand taking into account Eq.s(4.19), (4.20) (providing assurance that such anoperation indeed is legitimate and makes sense) we obtain,

∫

k∆

dxd∏

i=1

xi

1 − exp(−xi)exp (−f · x) =

∑

v∈V ertP

exp< f · v >[

d∏

i=1

(1 − exp−hvi (f)u

vi )]−1

=∑

x∈P∩Zd

exp< f · x > (4.23)

where the last equation was written in view of Eq.(1.11). From here, it is clearthat in the limit : f = 0 we reobtain back χ.

21To obtain correct results we needed to change signs in front of l1 and l2 . The same shouldbe done for other cones as well.

22Mathematically inclined reader is encoraged to read paper by Brion and Vergne, Ref.[76],where all missing details are scrupulously presented.

40

4.2 From Riemann-Roch-Hirzebruch to Witten and Lef-

schetz via Atiyah and Bott

As it was noticed already by Khovanskii and Puklikov [74] and elaborated byothers, e.g. see Ref.[76], in the limit : f = 0 the integral in the l.h.s of Eq.(4.21)can be associated with the Hirzebruch–Grotendieck- Riemann- Roch formulafor the Euler characteristic χ(E). In standard notations [76,77] it is given by

χ(E) =

∫

X

ch(E) · Td(TX), (4.24)

where E is a vector bundle over the varietyX , ch(E) is the Chern character of Eand Td(TX) is the Todd class of the tangent bundle TX of X . This formula istoo formal to be used immediately. The mathematical formalism of equivariantcohomology is needed for actual calculations connecting Eq.(4.22) with the l.h.sof Eq.(4.21). It was developed in the classical paper by Atiyah and Bott, Ref.[25]inspired by earlier work by Witten [26] on supersymmetry and Morse theory. Inthis work we shall use only a small portion of their results. A very pedagogicalexposition of the results by Atiyah and Bott can be found in the monograph byGuillemin and Sternberg [78] also containing helpful additional supersymmetricinformation.

We begin our discussion with the following observations. Earlier, in Section3.2 we introduced the Kirillov-Kostant symplectic two-form ωx. We noticed thatthis form is defined everywhere outside the set of critical points of symplecticmanifold M. The simplest example of the symplectic two- form was given inSection 3.4 for the case of two -sphere S2 where it coincides with the volumeform Ω = dφ ∧ dz for which

∫

S2 Ω = 4π. At the same time, the symplecticvolume form is given by Ω/2π so that the integral over S2 becomes equal to 2.This fact reminds us about the Gauss-Bonnet theorem for the 2-sphere which isprompting us to associate the two -form Ω/2π with the curvature two- form. Tomake things more interesting we recall some facts from the differential analysison complex manifolds as described, for example, in the book by Wells, Ref.[55].From this reference we find that the first Chern class c1(E) of the E vectorbundle over the sphere S2 is given by

c1(E) =i

π

dz ∧ dz(

1 + |z|2)2 =

2

π

ρdρdφ

(1 + ρ2)2 (4.25)

so that∫

S2 c1(E) = 2. Next, let us recall that any Kahler manifold is symplectic[61] and that for any Kahler manifold the second fundamental form Ω can be

written locally as Ω = i2

∑

ij hij(z)dzi∧dzj so that hij(z) = δij +O(|z|2). Hence,any symplectic volume form can be rewritten in terms of just described formΩ. The form Ω is closed but not exact. Evidently, up to a constant, c1(E) inEq.(5.2) coincides with the standard Kahler two -form. In view of the Gauss-Bonnet theorem, it is not exact. An easy calculation shows that the form dφ∧dz

41

can also be brought to the standard Kahler two- form (again up to a constant).Moreover, for the Hamiltonian of planar harmonic oscillator discussed in Section3.3. the standard symplectic two-form Ω can be written in several equivalentways

Ω = dx ∧ dy = rdr ∧ dθ =1

2dr2 ∧ dθ =

i

2dz ∧ dz (4.26)

and is certainly Kahlerian. For collection of k harmonic oscillators the symplec-tic two-form Ω is given, as usual, by Ω =

∑ki=1 dxi ∧ dyi = i

2

∑ki=1 dzi ∧ dzi so

that its n-th power is given by Ωn = Ω∧Ω∧ · · · ∧Ω =dx1 ∧ dy1 ∧ · · ·dxn ∧ dyn.In view of these results, it is convenient to introduce the differential form

exp Ω = 1 + Ω +1

2!Ω ∧ Ω +

1

3!Ω ∧ Ω ∧ Ω + · · · . (4.27)

By design, this expansion will have only k terms. Our earlier discussion of themoment map in Section 3.3. suggests that just described case of the collectionof harmonic oscillators is generic since its existence is guaranteed by the Morsetheory as discussed by Atiyah [62].23 In view of Eq.(4.23) such an expansioncan be formally associated with the total Chern class. Hence, we shall associateexpΩ with the total Chern class. Since all symplectic manifolds we consideredearlier possess singularities the standard homology and cohomology theoriesshould be replaced by equivariant ones as explained by Atiyah and Bott, Ref.[25].To this purpose we observe that in the absence of singularities the symplectic 2-form Ω is always closed, i.e. dΩ = 0. In case of singularities, one should replacethe exterior derivative d by d = d + i(ξ)24 while changing Ω to Ω − f · x innotations of of Eq.(4.21). The D-H integral, Eq.(4.21) can be formally rewrittennow as

∫

k∆

dx exp (−f · x) =

∫

k∆

exp(Ω) (4.28)

where Ω = Ω − f · x. The form Ω is eqivariantly closed. Indeed, since dΩ =dΩ+ i(ξ)Ω− f · dx then, in view of Eq.(4.1), i(ξ)Ω− f · dx = 0 by design, whiledΩ = 0 everywhere, except at singularities (critical points) where Ω = 0 (asdiscussed in Section 3.2.). Hence, dΩ = 0 as required. Since Ω can be identifiedwith the Chern class one should identify f · x with the Chern class as well, i.e.f · x ≡∑d

i=1 fici where we took into account that E = L1 ⊕ · · ·Ld becauseCd = C⊕ C ⊕ · · ·⊕ C so that Li is the line bundle associated with Ci . Aftersuch an identification Eq.(4.24) can be rewritten as

χ(E) =

∫

X

eΩ ·d∏

i=1

ci1 − exp(−ci)

. (4.29)

23In view of earlier discussed examples, we are interested only in the rotationally invariantobservables, this means that the θ (or φ) dependence in the two-form, Eq.(5.3), can bedropped which is equivalent to considering only the reduced phase space.This is meanigfulboth mathematically and physically. Details can be found in Ref.[33], pages 65-71.

24E.g. see Eq.(4.1).

42

Obtained result is in agreement with that given in the book by Guillemin,Ref.[38, p.60].

In view of the results of Part I, and Theorems 2.2. (by Solomon) and 2.5.(by Ginzburg) of Part II one can achieve more by discussing the intersectioncohomology ring of the reduced spaces associated with the D-H measures. Sincein Part I we noticed already that the Veneziano amplitudes can be formallyassociated with the period integrals for the Fermat (hyper)surfaces F and sincesuch integrals can be interpreted as intersection numbers between the cycleson F , one can formally rewrite the precursor to the Veneziano amplitude (asdiscussed in Part I) as

I =

(−∂∂f1

)r1

· · ·(−∂∂fd

)rd∫

∆

exp(Ω) |fi=0 ∀i=

∫

∆

dx(c1)r1 · · · (cd)rd (4.30)

provided that r1+···+rd = n . In such a language, the problem of calculation ofthe Veneziano amplitudes using generating function, Eq.(4.28), becomes math-ematically almost equivalent to earlier considered calculations related to theWitten-Kontsevich model discussed earlier in Ref.[42].THis circumstance willbe exploited in Part IV. Obtained results provide complete symplectic solutionof the Veneziano model.

As it was noticed by Atiyah and Bott [25], the replacement of exteriorderivative d by d = d + i(ξ) was inspired by earlier work by Witten on super-symmetric formulation of quantum mechanics and Morse theory, Ref.[26]. Suchan observation allows us to discuss calculation of χ and, hence, the Venezianoamplitudes using the supersymmetric formalism developed by Witten. Thetraditional way of developing Witten’s ideas is discussed in detail in earliermentioned monograph, Ref.[70]. Its essence is well summarized by Guillemin,Ref. [38]. Following this reference we notice that χ is equal to the dimensionQ = Q+−Q− of the quantum Hilbert space associated with the classical systemdescribed by the ( moment map) Hamiltonian as discussed earlier in this section. To describe quantum spaces associated with Q+ and Q− we need to remindour readers of several facts from the differential analysis on complex manifoldsdiscussed already in Part II..

We begin with the following observations. Let X be the complex Hermitianmanifold and let Ep+q(X) denote the complex -valued differential forms (sec-tions) of type (p, q) , p+ q = r, living on X . The Hodge decomposition insuresthat Er(X)=

∑

p+q=r Ep+q(X). The Dolbeault operators ∂ and ∂ act on Ep+q(X)

according to the rule ∂ : Ep+q(X) → Ep+1,q(X) and ∂ : Ep+q(X) → Ep,q+1(X), so that the exterior derivative operator is defined as d = ∂ + ∂. Let nowϕp,ψp ∈ Ep. By analogy with traditional quantum mechanics we define (usingDirac’s notations) the inner product

< ϕp | ψp >=

∫

M

ϕp ∧ ∗ψp (4.31)

where the bar means the complex conjugation and the star ∗ means the usualHodge conjugation. Use of such a product is motivated by the fact that the

43

period integrals, e.g. those for the Veneziano-like amplitudes, and, hence, thosegiven by Eq.(4.28), are expressible through such inner products [55]. Fortu-nately, such a product possesses properties typical for the finite dimensionalquantum mechanical Hilbert spaces. In particular,

< ϕp | ψq >= Cδp,q and < ϕp | ϕp >> 0, (4.32)

where C is some positive constant. With respect to such defined scalar product itis possible to define all conjugate operators, e.g. d∗, etc. and, most importantly,the Laplacians

∆ = dd∗ + d∗d,

= ∂∂∗ + ∂∗∂, (4.33)

= ∂∂∗ + ∂∗∂.

All this was known to mathematicians before Witten’s work [26]. The un-expected twist occurred when Witten suggested to extend the notion of theexterior derivative d. Within the de Rham picture (valid for both real and com-plex manifolds) let M be a compact Riemannian manifold and K be the Killingvector field which is just one of the generators of isometry of M, then Wittensuggested to replace the exterior derivative operator d by the extended operator

ds = d+ si(K) (4.34)

discussed earlier in the context of the equivariant cohomology. Here s is realnonzero parameter conveniently chosen. Witten argues that one can constructthe Laplacian (the Hamiltonian in his formulation) ∆ by replacing ∆ by ∆s =dsd

∗s + d∗sds . This is possible if and only if d2

s = d∗2s = 0 or, since d2s = sL(K) ,

where L(K) is the Lie derivative along the field K, if the Lie derivative actingon the corresponding differential form vanishes. The details are beautifully ex-plained in the much earlier paper by Frankel [68] mentioned earlier in Section 3.3.Atiyah and Bott observed that the auxiliary multicomponent parameter s can beidentified with earlier introduced f. This observation provides the link betweenthe symplectic D-H formalism discussed earlier and Witten’s supersymmetricquantum mechanics. Looking at Eq.s (4.31) and following Ref.s[3,38,39,70] we

consider the (Dirac) operator ∂ = ∂ + ∂∗ and its adjoint with respect to scalarproduct, Eq.(4.30), then use of the above references suggests that

Q = ker ∂ − co ker ∂∗ = Q+ −Q− = χ. (4.35)

in accord with Vergne[3]. The results just described provide yet another linkbetween the supersymmetric and symplectic formalisms. Additional details canbe found both in Part II and references just cited.

Note added in proof. After this work has been completed and accepted forpublication we become aware of the following two recent papers : arxiv:mathCo/0507163

44

and arxiv: mathCO/0504231.These papers are not only supporting results pre-sented in the main text, they also provide numerous additional details poten-tially useful for physical applications. Some of these applications will be dis-cussed in Part IV.

45

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